An Infinite Dimensional Umbral Calculus

Home , Appell sequence, Binomial type, Delta operator, Polynomial sequence, Sheffer sequence

An infinite dimensional umbral calculus Dmitri Finkelshtein Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.; e-mail: [email protected] Yuri Kondratiev FakultätfürMathematik, UniversitätBielefeld, 33615 Bielefeld, Germany; e-mail: [email protected] Eugene Lytvynov Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.; e-mail: [email protected] Maria JoãoOliveira Departamento de Ciênciase Tecnologia, Universidade Aberta, 1269-001 Lisbon, Por- tugal; CMAF-CIO, University of Lisbon, 1749-016 Lisbon, Portugal; e-mail: [email protected]

Abstract

The aim of this paper is to develop foundations of umbral calculus on the space D0 of distri- d 0 butions on R , which leads to a general theory of Sheffer polynomial sequences on D . We define a sequence of monic polynomials on D0, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D0 to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on R of binomial type to a polynomial sequence of 0 0 binomial type on D , and a lifting of a Sheffer sequence on R to a Sheffer sequence on D . Examples of lifted polynomial sequences include the falling and rising factorials on D0, Abel, Hermite, Charlier, and Laguerre polynomials on D0. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role. arXiv:1701.04326v2 [math.FA] 5 Mar 2018 Keywords: Generating function; polynomial sequence on D0; polynomial sequence of binomial type on D0; Sheffer sequence on D0; shift-invariance; umbral calculus on D0. 2010 MSC. Primary: 05A40, 46E50. Secondary: 60H40, 60G55.

1 Introduction

In its modern form, umbral calculus is a study of shift-invariant linear operators acting on polynomials, their associated polynomial sequences of binomial type, and Sheﬀer sequences (including Appell sequences). We refer to the seminal papers [29, 38, 39],

1 see also the monographs [24, 37]. Umbral calculus has applications in combinatorics, theory of special functions, approximation theory, probability and statistics, topology, and physics, see e.g. the survey paper [12] for a long list of references. Many extensions of umbral calculus to the case of polynomials of several, or even infinitely many variables were discussed e.g. in [5,10,14,28,32,35,36,41,42], for a longer list of such papers see the introduction to [13]. Appell and Sheffer sequences of polynomials of several noncommutative variables arising in the context of free probability, Boolean probability, and conditionally free probability were discussed in [2–4], see also the references therein. The paper [13] was a pioneering (and seemingly unique) work in which elements of basis-free umbral calculus were developed on an infinite dimensional space, more precisely, on a real separable Hilbert space H. This paper discussed, in particular, shift- invariant linear operators acting on the space of polynomials on H, Appell sequences, and examples of polynomial sequences of binomial type. In fact, examples of Sheffer sequences, i.e., polynomial sequences with generating function of a certain exponential type, have appeared in infinite dimensional analysis on numerous occasions. Some of these polynomial sequences are orthogonal with respect to a given probability measure on an infinite dimensional space, while others are related to analytical structures on such spaces. Typically, these polynomials are either defined on a co-nuclear space Φ0 (i.e, the dual of a nuclear space Φ), or on an appropriate subset of Φ0. Furthermore, in majority of examples, the nuclear space Φ consists of (smooth) functions on an underlying space X. For simplicity, we choose to work in this paper with the Gel’fand triple

2 d 0 0 Φ = D ⊂ L (R , dx) ⊂ D = Φ .

Here D is the nuclear space of smooth compactly supported functions on Rd, d ∈ N, and D0 is the dual space of D, where the dual pairing between D0 and D is obtained by continuously extending the inner product in L2(Rd, dx). Let us mention several known examples of Sheﬀer sequences on D0 or its subsets:

(i) In inﬁnite dimensional Gaussian analysis, also called white noise analysis, Her- mite polynomial sequences on D0 (or rather on S0 ⊂ D0, the Schwartz space of tempered distributions) appear as polynomials orthogonal with respect to Gaus- sian white noise measure, see e.g. [6,15,16,31]. (ii) Charlier polynomial sequences on the conﬁguration space of counting Radon measures on Rd,Γ ⊂ D0, appear as polynomials orthogonal with respect to Poisson point process on Rd, see [17,19,22]. (iii) Laguerre polynomial sequences on the cone of discrete Radon measures on Rd, K ⊂ D0, appear as polynomials orthogonal with respect to the gamma random measure, see [21,22].

2 (iv) Meixner polynomial sequences on D0 appear as polynomials orthogonal with respect to the Meixner white noise measure, see [25,26].

(v) Special polynomials on the configuration space Γ ⊂ D0 are used to construct the K-transform, see e.g. [7,18,20]. Recall that the K-transform determines the duality between point processes on Rd and their correlation measures. These polynomials will be identified in this paper as the infinite dimensional analog of the falling factorials (a special case of the Newton polynomials).

(vi) Polynomial sequences on D0 with generating function of a certain exponential type are used in biorthogonal analysis related to general measures on D0, see [1,23].

Note, however, that even the very notion of a general polynomial sequence on an infinite dimensional space has never been discussed! The classical umbral calculus on the real line gives a general theory of Sheffer sequences and related (umbral) operators. So our aim in this paper is to develop foundations of umbral calculus on the space D0, which will eventually lead to a general theory of Sheffer sequences on D0 and their umbral operators. In fact, at a structural level, many results of this paper will have remarkable similarities to the classical setting of polynomials on R. For example, the form of the generating function of a Sheffer sequence on D0 will be similar to the generating function of a Sheffer sequence on R: the constants appearing in the latter function are replaced in the former function by appropriate linear continuous operators. There is a principal point in our approach that we would like to stress. The paper [13] deals with polynomials on a general Hilbert space H, while the monograph [6] develops Gaussian analysis on a general co-nuclear space Φ0, without the assumption that Φ0 consists of generalized functions on Rd (or on a general underlying space). In fact, we will discuss in Remark 7.5 below that the case of the infinite dimensional Hermite polynomials is, in a sense, exceptional and does not require from the co-nuclear space Φ0 any special structure. In all other cases, the choice Φ0 = D0 is crucial. Having said this, let us note that our ansatz can still be applied to a rather general co-nuclear space of generalized functions over a topological space X, equipped with a reference measure. We also stress that the topological aspects of the spaces D, D0, and their symmetric tensor powers are crucial for us since all the linear operators (appearing as ‘coefficients’ in our theory) are continuous in the respective topologies. Furthermore, this assumption of continuity is principal and cannot be omitted. The origins of the classical umbral calculus are in combinatorics. So, by analogy, one can think of umbral calculus on D0 as a kind of spatial combinatorics. To give the reader P∞ a better feeling of this, let us consider the following example. Let γ = i=1 δxi ∈ Γ be a configuration. Here δxi denotes the Dirac measure with mass at xi. We will construct (the kernel of) the falling factorial, denoted by (γ)n, as a function from Γ to

3 D0 n. (Here and below denotes the symmetric tensor product.) This will allow us γ 1 to deﬁne ‘γ choose n’ by n := n! (γ)n. And we will get the following explicit formula which supports this term:

γ X = δx δx · · · δx , (1.1) n i1 i2 in {i1,...,in}⊂N

i.e., the sum is obtained by choosing all possible n-point subsets from the (locally ﬁnite)

set {xi}i∈N. The latter set can be obviously identified with the configuration γ. The paper is organized as follows. In Section 2 we discuss preliminaries. In partic- 0 ular, we recall the construction of a general Gel’fand triple Φ ⊂ H0 ⊂ Φ , where Φ is a nuclear space and Φ0 is the dual of Φ (a co-nuclear space) with respect to the center 0 0 Hilbert space H0. We consider the space P(Φ ) of polynomials on Φ and equip it with a nuclear space topology. Its dual space, denoted by F(Φ0), has a natural (commutative) algebraic structure with respect to the symmetric tensor product. We also define a family of shift operators, (E(ζ))ζ∈Φ0 , and the space of shift-invariant continuous linear operators on P(Φ0), denoted by S(P(Φ0)). In Section 3, we give the definitions of a polynomial sequence on Φ0, a monic polynomial sequence on Φ0, and a monic polynomial sequence on Φ0 of binomial type. Starting from Section 4, we choose the Gel’fand triple as D ⊂ L2(Rd, dx) ⊂ D0. The main result of this section is Theorem 4.1, which gives three equivalent conditions for a monic polynomial sequence to be of binomial type. The first equivalent condition is that the corresponding lowering operators are shift-invariant. The second condition gives a representation of each lowering operator through directional derivatives in directions d of delta functions, δx (x ∈ R ). The third condition gives the form of the generating function of a polynomial sequence of binomial type. To prove Theorem 4.1, we derive two essential results. The first one is an operator expansion theorem (Theorem 4.7), which gives a description of any shift-invariant 0 operator T ∈ S(P(D )) in terms of the lowering operators in directions δx. The second result is an isomorphism theorem (Theorem 4.9): we construct a bijection J between the spaces S(P(D0)) and F(D0) such that, under J, the product of any shift-invariant operators goes over into the symmetric tensor product of their images. This implies, in particular, that any two shift-invariant operators commute. Next, we define a family of delta operators on D0 and prove that, for each such family, there exists a unique monic polynomial sequence of binomial type for which these delta operators are the lowering operators. In Section 5, we identify a procedure of the lifting of a polynomial sequence of binomial type on R to a polynomial sequence of binomial type on D0. This becomes possible due to the structural similarities between the one-dimensional and infinite- dimensional theories. Using this procedure, we identify, on D0, the falling factorials, the rising factorials, the Abel polynomials, and the Laguerre polynomials of binomial

4 type. We stress that the polynomial sequences lifted from R to D0 form a subset of a (much larger) set of all polynomial sequences of binomial type on D0. In Section 6, we define a Sheffer sequence on D0 as a monic polynomial sequence on D0 whose lowering operators are delta operators. Thus, to every Sheffer sequence, there corresponds a (unique) polynomial sequence of binomial type. In particular, if the corresponding binomial sequence is just the set of monomials (i.e., their delta operators are differential operators), we call such a Sheffer sequence an Appell sequence. The main result of this section, Theorem 6.2, gives several equivalent conditions for a monic polynomial sequence to be a Sheffer sequence. In particular, we find the generating function of a Sheffer sequence on D0. In Section 7, we extend the procedure of the lifting described in Section 5 to Sheffer sequences. Thus, for each Sheffer sequence on R, we define a Sheffer sequence on D0. Using this procedure, we recover, in particular, the Hermite polynomials, the Charlier polynomials, and the orthogonal Laguerre polynomials on D0. Finally, in Appendix, we discuss several properties of formal tensor power series. From the technical point of view, the similarities between the infinite dimensional and the classical settings open new perspectives in infinite dimensional analysis. Due to the special character of this approach, namely, through definition of umbral operators on P(D0) and umbral composition of polynomials on D0, further developments and applications in infinite dimensional analysis are subject of forthcoming publications. Let us also mention the open problem of (at least partial) characterization of Sheffer sequences that are orthogonal with respect to a certain probability measure on D0. In the one-dimensional case, such a characterization is due to Meixner [27]. For multi- dimensional extensions of this result, see [11,33,34] and the references therein.

2 Preliminaries

2.1 Nuclear and co-nuclear spaces Let us ﬁrst recall the deﬁnition of a nuclear space, for details see e.g. [8, Chapter 14, Section 2.2]. Consider a family of real separable Hilbert spaces (Hτ )τ∈T , where T is T an arbitrary indexing set. Assume that the set Φ := τ∈T Hτ is dense in each Hilbert space Hτ and the family (Hτ )τ∈T is directed by embedding, i.e., for any τ1, τ2 ∈ T there exists a τ3 ∈ T such that Hτ3 ⊂ Hτ1 and Hτ3 ⊂ Hτ2 and both embeddings are continuous. We introduce in Φ the projective limit topology of the Hτ spaces:

Φ = proj lim Hτ . τ∈T

By deﬁnition, the sets {ϕ ∈ Φ | kϕ − ψkHτ < ε} with ψ ∈ Φ, τ ∈ T , and ε > 0 form a system of base neighborhoods in this topology. Here k · kHτ denotes the norm in Hτ .

5 Assume that, for each τ1 ∈ T , there exists a τ2 ∈ T such that Hτ2 ⊂ Hτ1 , and the

operator of embedding of Hτ2 into Hτ1 is of the Hilbert–Schmidt class. Then the linear topological space Φ is called nuclear. Next, let us assume that, for some τ0 ∈ T , each Hilbert space Hτ with τ ∈ T is continuously embedded into H0 := Hτ0 . We will call H0 the center space. 0 Let Φ denote the dual space of Φ with respect to the center space H0, i.e., the dual pairing between Φ0 and Φ is obtained by continuously extending the inner product in 0 H0, see e.g. [8, Chapter 14, Section 2.3]. The space Φ is often called co-nuclear. 0 S By the Schwartz theorem (e.g. [8, Chapter 14, Theorem 2.1]), Φ = τ∈T H−τ , where H−τ denotes the dual space of Hτ with respect to the center space H0. We endow Φ0 with the Mackey topology—the strongest topology in Φ0 consistent with the duality between Φ and Φ0 (i.e., the set of continuous linear functionals on Φ0 coincides with Φ). The Mackey topology in Φ0 coincides with the topology of the inductive limit of the H−τ spaces, see e.g. [40, Chapter IV, Proposition 4.4] or [6, Chapter 1, Section 1]. Thus, we obtain the Gel’fand triple (also called the standard triple)

0 Φ = proj lim Hτ ⊂ H0 ⊂ ind lim H−τ = Φ . (2.1) τ∈T τ∈T Let X and Y be linear topological spaces that are locally convex and Hausdorﬀ. (Both Φ and Φ0 are such spaces.) We denote by L(X,Y ) the space of continuous linear operators acting from X into Y . We will also denote L(X) := L(X,X). We denote by X0 and Y 0 the dual space of X and Y , respectively. We endow X0 with the Mackey topology with respect to the duality between X and X0. We similarly endow Y 0 with the Mackey topology. Each operator A ∈ L(X,Y ) has the adjoint operator A∗ ∈ L(Y 0,X0) (also called the transpose of A or the dual of A), see e.g. [30, Theorem 8.11.3]. Remark 2.1. Note that, since we chose the Mackey topology on Φ0, for an operator A ∈ L(Φ0), we have A∗ ∈ L(Φ). This fact will be used throughout the paper.

Proposition 2.2. Consider the Gel’fand triple (2.1). Let A :Φ → Φ and B :Φ0 → Φ0 be linear operators. (i) We have A ∈ L(Φ) if and only if, for each τ1 ∈ T , there exists a τ2 ∈ T such ˆ that the operator A can be extended by continuity to an operator A ∈ L(Hτ2 , Hτ1 ). 0 (ii) We have B ∈ L(Φ ) if and only if, for each τ1 ∈ T , there exists a τ2 ∈ T such ˆ ˆ that the operator B := B H−τ1 takes on values in H−τ2 and B ∈ L(H−τ1 , H−τ2 ). Remark 2.3. Proposition 2.2 admits a sraightforward generalization to the case of two 0 0 Gel’fand triples, Φ ⊂ H0 ⊂ Φ and Ψ ⊂ G0 ⊂ Ψ , and linear operators A :Φ → Ψ and B :Φ0 → Ψ0. Remark 2.4. Part (ii) of Proposition 2.2 is related to the universal property of an inductive limit, which states that any linear operator from an inductive limit of a

6 family of locally convex spaces to another locally convex space is continuous if and only if the restriction of the operator to any member of the family is continuous, see e.g. [9, II.29]. Proof of Proposition 2.2. (i) By the deﬁnition of the topology in Φ, the linear operator A :Φ → Φ is continuous if and only if, for any τ1 ∈ T and ε1 > 0, there exist τ2 ∈ T and ε2 > 0 such that the pre-image of the set {ϕ ∈ Φ | kϕk < ε } Hτ1 1 contains the set {ϕ ∈ Φ | kϕk < ε }. Hτ2 2 But this implies the statement. (ii) Assume B ∈ L(Φ0). Then, by Remark 2.1, we have B∗ ∈ L(Φ). Hence, for each ∗ τ1 ∈ T , there exists a τ2 ∈ T such that the operator B can be extended by continuity ˆ∗ ˆ∗ ˆ to an operator B ∈ L(Hτ2 , Hτ1 ). But the adjoint of the operator B is B := B H−τ1 . ˆ Hence B ∈ L(H−τ1 , H−τ2 ). Conversely, assume that, for each τ1 ∈ T , there exists a τ2 ∈ T such that the ˆ ˆ operator B := B H−τ1 takes on values in H−τ2 and B ∈ L(H−τ1 , H−τ2 ). Therefore, ˆ∗ ˆ∗ B ∈ L(Hτ2 , Hτ1 ). Denote A := B Φ. As easily seen, the deﬁnition of the operator A does not depend on the choice of τ1, τ2 ∈ T . Hence, A :Φ → Φ, and by part (i) we conclude that A ∈ L(Φ). But B = A∗ and hence B ∈ L(Φ0). In what follows, ⊗ will denote the tensor product. In particular, for a real separable ⊗n Hilbert space H, H denotes the nth tensor power of H. We will denote by Symn ∈ L(H⊗n) the symmetrization operator, i.e., the orthogonal projection satisfying 1 X Sym f ⊗ f ⊗ · · · ⊗ f = f ⊗ f ⊗ · · · ⊗ f (2.2) n 1 2 n n! σ(1) σ(2) σ(n) σ∈S(n) for f1, f2, . . . , fn ∈ H. Here S(n) denotes the symmetric group acting on {1, . . . , n}. We will denote the symmetric tensor product by . In particular,

f1 f2 · · · fn := Symn f1 ⊗ f2 ⊗ · · · ⊗ fn, f1, f2, . . . , fn ∈ H,

n ⊗n and H := Symn H is the nth symmetric tensor power of H. Note that, for each f ∈ H, we have f n = f ⊗n. Starting with Gel’fand triple (2.1), one constructs its nth symmetric tensor power as follows: n n n n 0 n Φ := proj lim Hτ ⊂ H0 ⊂ ind lim H−τ =: Φ , τ∈T τ∈T see e.g. [6, Section 2.1] for details. In particular, Φ n is a nuclear space and Φ0 n is its n 0 0 0 0 dual with respect to the center space H0 . We will also denote Φ = H0 = Φ := R. The dual pairing between F (n) ∈ Φ0 n and g(n) ∈ Φ n will be denoted by hF (n), g(n)i.

7 Remark 2.5. Consider the set {ξ⊗n | ξ ∈ Φ}. By the polarization identity, the linear n span of this set is dense in every space Hτ , τ ∈ T . The following lemma will be very important for our considerations.

Lemma 2.6. (i) Let F (n),G(n) ∈ Φ0 n be such that

hF (n), ξ⊗ni = hG(n), ξ⊗ni for all ξ ∈ Φ,

then F (n) = G(n). (ii) Let Φ and Ψ be nuclear spaces and let A, B ∈ L(Φ n, Ψ). Assume that

Aξ⊗n = Bξ⊗n for all ξ ∈ Φ.

Then A = B.

Proof. Statement (i) follows from Remark 2.5, statement (ii) follows from Proposi- tion 2.2, (i) and Remarks 2.3 and 2.5.

2.2 Polynomials on a co-nuclear space Below we ﬁx the Gel’fand triple (2.1). Deﬁnition 2.7. A function P :Φ0 → R is called a polynomial on Φ0 if

n X P (ω) = hω⊗k, f (k)i, ω ∈ Φ0, (2.3) k=0

(k) k ⊗0 (n) where f ∈ Φ , k = 0, 1, . . . , n, n ∈ N0 := {0, 1, 2,... }, and ω := 1. If f 6= 0, one says that the polynomial P is of degree n. We denote by P(Φ0) the set of all polynomials on Φ0. Remark 2.8. For each P ∈ P(Φ0), its representation in form (2.3) is evidently unique. (k) k (n) n For any f ∈ Φ and g ∈ Φ , k, n ∈ N0, we have hω⊗k, f (k)ihω⊗n, g(n)i = hω⊗(k+n), f (k) g(n)i, ω ∈ Φ0. (2.4)

Hence P(Φ0) is an algebra under point-wise multiplication of polynomials on Φ0. 0 We will now define a topology on P(Φ ). Let Ffin(Φ) denote the topological direct n sum of the nuclear spaces Φ , n ∈ N0. Hence, Ffin(Φ) is a nuclear space, see e.g. [6, Section 5.1]. This space consists of all finite sequences f = (f (0), f (1), . . . , f (n), 0, 0,... ), (k) k where f ∈ Φ , k = 0, 1, . . . , n, n ∈ N0. The convergence in Ffin(Φ) means the uniform finiteness of non-zero elements and the coordinate-wise convergence in each Φ k.

8 Remark 2.9. Below we will often identify f (n) ∈ Φ n with

(n) (0,..., 0, f , 0, 0,... ) ∈ Fﬁn(Φ).

0 We deﬁne a natural bijective mapping I : Fﬁn(Φ) → P(Φ ) by n X (If)(ω) := hω⊗k, f (k)i, (2.5) k=0

(0) (1) (n) for f = (f , f , . . . , f , 0, 0,... ) ∈ Ffin(Φ). We define a nuclear space topology on 0 P(Φ ) as the image of the topology on Ffin(Φ) under the mapping I. The space Ffin(Φ) may be endowed with the structure of an algebra with respect to the symmetric tensor product

k !∞ X f g = f (i) g(k−i) , (2.6) i=0 k=0 (k) ∞ (k) ∞ where f = (f )k=0, g = (g )k=0 ∈ Ffin(Φ). The unit element of this (commutative) algebra is the vacuum vector Ω := (1, 0, 0 ...). By (2.4)–(2.6), the bijective mapping I provides an isomorphism between the alge- 0 bras Ffin(Φ) and P(Φ ), namely, for any f, g ∈ Ffin(Φ), I(f g)(ω) = (If)(ω)(Ig)(ω), ω ∈ Φ0. Let ∞ Y F(Φ0) := Φ0 k k=0 denote the topological product of the spaces Φ0 k. The space F(Φ0) consists of all (k) ∞ (k) 0 k sequences F = (F )k=0, where F ∈ Φ , k ∈ N0. Note that the convergence in this space means the coordinate-wise convergence in each space Φ0 k. (k) ∞ 0 Each element F = (F )k=0 ∈ F(Φ ) determines a continuous linear functional on Ffin(Φ) by ∞ X (k) (k) (k) ∞ hF, fi := hF , f i, f = (f )k=0 ∈ Ffin(Φ) (2.7) k=0 0 (note that the sum in (2.7) is, in fact, finite). The dual of Ffin(Φ) is equal to F(Φ ), and the topology on F(Φ0) coincides with the Mackey topology on F(Φ0) that is consistent 0 with the duality between Ffin(Φ) and F(Φ ), see e.g. [6]. In view of the definition of the topology on P(Φ0), we may also think of F(Φ0) as the dual space of P(Φ0). Similarly to (2.6), one can introduce the symmetric tensor product on F(Φ0):

k !∞ X F G = F (i) G(k−i) , (2.8) i=0 k=0

9 (k) ∞ (k) ∞ 0 where F = (F )k=0,G = (G )k=0 ∈ F(Φ ). The unit element of this algebra is again Ω = (1, 0, 0,...). We will now discuss another realization of the space F(Φ0). We denote by S(R, R) P∞ n the vector space of formal series R(t) = n=0 rnt in powers of t ∈ R, where rn ∈ R for n ∈ N0. The S(R, R) is an algebra under the product of formal power series. Similarly to S(R, R), we give the following (n) ∞ 0 Definition 2.10. Each (F )n=0 ∈ F(Φ ) identifies a ‘real-valued’ formal series P∞ (n) ⊗n n=0hF , ξ i in tensor powers of ξ ∈ Φ. We denote by S(Φ, R) the vector space of such formal series with natural operations. We define a product on S(Φ, R) by

∞ ! ∞ ! ∞ * n + X X X X hF (n), ξ⊗ni hG(n), ξ⊗ni = F (i) G(n−i), ξ⊗n , (2.9) n=0 n=0 n=0 i=0

(n) ∞ (n) ∞ 0 where (F )n=0, (G )n=0 ∈ F(Φ ). (n) ∞ (n) ∞ 0 Remark 2.11. Assume that, for some (F )n=0, (G )n=0 ∈ F(Φ ) and ξ ∈ Φ, both P∞ (n) ⊗n P∞ (n) ⊗n series n=0hF , ξ i and n=0hG , ξ i converge absolutely. Then also the series on the right hand side of (2.9) converges absolutely and (2.9) holds as an equality of two real numbers. (n) ∞ 0 Remark 2.12. Let t ∈ R and ξ ∈ Φ. Then, tξ ∈ Φ and for (F )n=0 ∈ F(Φ ),

∞ ∞ X X hF (n), (tξ)⊗ni = tnhF (n), ξ⊗ni, (2.10) n=0 n=0 the expression on the right hand side of equality (2.10) being the formal power series in t that has coeﬃcient hF (n), ξ⊗ni by tn. According to the deﬁnition of S(Φ, R), there exists a natural bijective mapping I : F(Φ0) → S(Φ, R) given by

∞ X (n) ⊗n (n) ∞ 0 (IF )(ξ) := hF , ξ i,F = (F )n=0 ∈ F(Φ ), ξ ∈ Φ. (2.11) n=0

The mapping I provides an isomorphism between the algebras F(Φ0) and S(Φ, R), namely, for any F,G ∈ F(Φ0),

I(F G)(ξ) = (IF )(ξ)(IG)(ξ) (2.12) see (2.8) and (2.9). Remark 2.13. In view of the isomorphism I, we may think of S(Φ, R) as the dual space of P(Φ0).

10 Analogously to Definition 2.10 and Remark 2.12, we can introduce a space of Φ- valued tensor power series. ∞ n Definition 2.14. Let (An)n=1 be a sequence of operators An ∈ L(Φ , Φ). Then the ∞ P∞ ⊗n operators (An)n=1 can be identified with a ‘Φ-valued’ formal series n=1 Anξ in tensor powers of ξ ∈ Φ. We denote by S(Φ, Φ) the vector space of such formal series. ∞ Remark 2.15. Let t ∈ R and ξ ∈ Φ. Then, tξ ∈ Φ and for a sequence (An)n=1 as in Definition 2.14 ∞ ∞ X ⊗n X n ⊗n An(tξ) = t Anξ n=1 n=1 ⊗n n is the formal power series in t that has coefficient Anξ ∈ Φ by t . Recall that, by ⊗n n Lemma 2.6, (ii), the values of the operator An on the vectors ξ ∈ Φ uniquely identify the operator An. In Appendix, we discuss several properties of formal tensor power series.

2.3 Shift-invariant operators Definition 2.16. For each ζ ∈ Φ0, we define the operator D(ζ): P(Φ0) → P(Φ0) of differentiation in direction ζ by

P (ω + tζ) − P (ω) (D(ζ)P )(ω) := lim ,P ∈ P(Φ0), ω ∈ Φ0. t→0 t

0 Definition 2.17. For each ζ ∈ Φ , we define the annihilation operator A(ζ) ∈ L(Ffin(Φ)) by ⊗n ⊗(n−1) A(ζ)Ω := 0, A(ζ)ξ := nhζ, ξiξ for ξ ∈ Φ, n ∈ N. Lemma 2.18. For each ζ ∈ Φ0, we have D(ζ) ∈ L(P(Φ0)).

0 Proof. Using the bijection I : Fﬁn(Φ) → P(Φ ) deﬁned by (2.5), we easily obtain

D(ζ) = IA(ζ)I−1, ζ ∈ Φ0,

which implies the statement. Deﬁnition 2.19. For each ζ ∈ Φ0, we deﬁne the operator E(ζ): P(Φ0) → P(Φ0) of shift by ζ by (E(ζ)P )(ω) := P (ω + ζ),P ∈ P(Φ0), ω ∈ Φ0.

Lemma 2.20. (Boole’s formula) For each ζ ∈ Φ0,

∞ X 1 E(ζ) = D(ζ)k. k! k=0

11 Proof. Note that the infinite sum is, in fact, a finite sum when applied to a polynomial, and thus it is a well-defined operator on P(Φ0). For each ξ ∈ Φ and n ∈ N,

n X n E(ζ)h·⊗n, ξ⊗ni(ω) = h(ω + ζ)⊗n, ξ⊗ni = hω, ξikhζ, ξin−k k k=0 n ∞ X 1 X 1 = D(ζ)kh·⊗n, ξ⊗ni(ω) = D(ζ)kh·⊗n, ξ⊗ni(ω), k! k! k=0 k=0 which implies the statement. Note that Lemmas 2.18 and 2.20 imply that E(ζ) ∈ L(P(Φ0)) for each ζ ∈ Φ0. Deﬁnition 2.21. We say that an operator T ∈ L(P(Φ0)) is shift-invariant if

TE(ζ) = E(ζ)T for all ζ ∈ Φ0.

We denote the linear space of all shift-invariant operators by S(P(Φ0)). The space S(P(Φ0)) is an algebra under the usual product (composition) of operators. Obviously, for each ζ ∈ Φ0, the operators D(ζ) and E(ζ) belong to S(P(Φ0)).

3 Monic polynomial sequences on a co-nuclear space

We will now introduce the notion of a polynomial sequence on Φ0. (n) 0 0 For each n ∈ N0, we denote by P (Φ ) the subspace of P(Φ ) that consists of all polynomials on Φ0 of degree ≤ n.

Lemma 3.1. A mapping P(n) :Φ n → P(n)(Φ0) is linear and continuous if and only if it is of the form (n) (n) (n) (n) P f (ω) = hP (ω), f i, (3.1) where P (n) :Φ0 → Φ0 n is a continuous mapping of the form

n (n) X ⊗k P (ω) = Un,k ω (3.2) k=0

0 k 0 n with Un,k ∈ L(Φ , Φ ).

(n) n (n) 0 (n) 0 Proof. Let P ∈ L Φ , P (Φ ) . Then, by the deﬁnition of P (Φ ), there exist n k (n) n 0 operators Vk,n ∈ L(Φ , Φ ), k = 0, 1, . . . , n, such that, for any f ∈ Φ and ω ∈ Φ

n (n) (n) X ⊗k (n) P f (ω) = hω ,Vk,nf i k=0

12 n X ⊗k (n) = hUn,k ω , f i, k=0

∗ 0 k 0 n (n) where Un,k := Vk,n ∈ L(Φ , Φ ). Conversely, every P of the form (3.2) determines (n) n (n) 0 P ∈ L Φ , P (Φ ) by formula (3.1).

(n) 0 0 n Deﬁnition 3.2. Assume that, for each n ∈ N0, P :Φ → Φ is of the form (3.2) 0 k 0 n 0 n with Un,k ∈ L(Φ , Φ ). Furthermore, assume that, for each n ∈ N0, Un,n ∈ L(Φ ) (n) ∞ 0 is a homeomorphism. Then we call (P )n=0 a polynomial sequence on Φ . 0 n If additionally, for each n ∈ N0, Un,n = 1, the identity operator on Φ , then we (n) ∞ 0 call (P )n=0 a monic polynomial sequence on Φ . Remark 3.3. Below, to simplify notations, we will only deal with monic polynomial sequences. The results of this paper can be extended to the case of a general polynomial sequence on Φ0 = D0. Remark 3.4. By the deﬁnition of a monic polynomial sequence we get

n−1 (n) (n) ⊗n (n) X ⊗k (n) (n) n hP (ω), f i = hω , f i + hω ,Vk,nf i, f ∈ Φ (3.3) k=0

∗ n k where Vk,n := Un,k ∈ L(Φ , Φ ).

(n) ∞ 0 Lemma 3.5. Let (P )n=0 be a monic polynomial sequence on Φ . The following statements hold. n k (i) There exist operators Rk,n ∈ L(Φ , Φ ), k = 0, 1, . . . , n − 1, n ∈ N, such that, for all ω ∈ Φ0 and f (n) ∈ Φ n,

n−1 ⊗n (n) (n) (n) X (k) (n) hω , f i = hP (ω), f i + hP (ω),Rk,nf i. (3.4) k=0 (ii) We have

( n ) 0 X (k) (k) (k) k P(Φ ) = hP , f i | f ∈ Φ , k = 0, 1, . . . , n, n ∈ N0 . (3.5) k=0 Proof. (i) We prove by induction on n. For n = 1, the statement trivially holds. Assume that the statement holds for 1, 2, . . . , n. Then, by using (3.3) and the induction assumption, we get

n ⊗(n+1) (n+1) (n+1) (n+1) X ⊗k (n+1) hω , f i = hP (ω), f i − hω ,Vk,n+1f i k=0

13 n k−1 ! (n+1) (n+1) X (k) (n+1) X (i) (n+1) = hP (ω), f i − hP (ω),Vk,n+1f i + hP (ω),Ri,kVk,n+1f i , k=0 i=0 which implies the statement for n + 1. (ii) This follows immediately from (i).

(n) ∞ 0 0 Deﬁnition 3.6. Let (P )n=0 be a monic polynomial sequence on Φ . For each ζ ∈ Φ , we deﬁne a lowering operator Q(ζ) as the linear operator on P(Φ0) (cf. (3.5)) satisfying

(n) (n) (n−1) (n) (n) n Q(ζ)hP , f i := hP , A(ζ)f i, f ∈ Φ , n ∈ N, (0) (0) (0) Q(ζ)hP , f i := 0, f ∈ R, where the operator A(ζ) is deﬁned by Deﬁnition 2.17. Lemma 3.7. For every ζ ∈ Φ0, we have Q(ζ) ∈ L(P(Φ0)).

(n) n Proof. We deﬁne an operator R ∈ L(Fﬁn(Φ)) by setting, for each f ∈ Φ ,  R f (n), k < n  k,n Rf (n)(k) := f (n), k = n, 0, k > n,

where the operators Rk,n are as in (3.4). Similarly, using the operators Vk,n from formula (3.3), we define an operator V ∈ L(Ffin(Φ)). As easily seen, Q(ζ) = IV A(ζ)RI−1, where I is the homeomorphism defined by (2.5). This implies the required result. The simplest example of a monic polynomial sequence on Φ0 is P (n)(ω) = ω⊗n, (n) n (n) (n) ⊗n (n) n ∈ N0. In this case, for f ∈ Φ , hP (ω), f i = hω , f i is just a monomial on Φ0 of degree n. For each ζ ∈ Φ0, we obviously have Q(ζ) = D(ζ), i.e., the corresponding lowering operators are just differentiation operators. Furthermore, we trivially see in this case that, for any n ∈ N and any ω, ζ ∈ Φ0, n X n P (n)(ω + ζ) = P (k)(ω) P (n−k)(ζ). (3.6) k k=0

(n) ∞ 0 Deﬁnition 3.8. Let (P )n=0 be a monic polynomial sequence on Φ . We say that (n) ∞ 0 (P )n=0 is of binomial type if, for any n ∈ N and any ω, ζ ∈ Φ , formula (3.6) holds. (n) ∞ Remark 3.9. A monic polynomial sequence (P )n=0 is of binomial type if and only if, for any n ∈ N, ω, ζ ∈ Φ0, and ξ ∈ Φ, n X n hP (n)(ω + ζ), ξ⊗ni = hP (k)(ω), ξ⊗kihP (n−k)(ζ), ξ⊗(n−k)i. k k=0

14 The following lemma will be important for our considerations.

(n) ∞ 0 Lemma 3.10. Let (P )n=0 be a monic polynomial sequence on Φ of binomial type. Then, for each n ∈ N, P (n)(0) = 0. Proof. We proceed by induction on n. For n = 1, it follows from (3.6) that P (1)(ω + ζ) = P (1)(ω) + P (1)(ζ), ω, ζ ∈ Φ0. Setting ζ = 0, one obtains P (1)(0) = 0. Assume that the statement holds for 1, 2, . . . n. Then, for all ω, ζ ∈ Φ0, n X n + 1 P (n+1)(ω + ζ) = P (n+1)(ω) + P (k)(ω) P (n+1−k)(ζ) + P (n+1)(ζ). k k=1 Setting ζ = 0, we conclude P (n+1)(0) = 0.

4 Equivalent characterizations of a polynomial sequence of binomial type

Our next aim is to derive equivalent characterizations of a polynomial sequence on Φ0 of binomial type. As mentioned in Introduction, it will be important for our considerations that Φ0 will be chosen as a space of generalized functions. ∞ d So we ﬁx d ∈ N and choose Φ to be the nuclear space D := C0 (R ) of all real-valued smooth functions on Rd with compact support. More precisely, let T denote the set ∞ d d of all pairs (l, ϕ) with l ∈ N0 and ϕ ∈ C (R ), ϕ(x) ≥ 1 for all x ∈ R . For each l,2 d τ = (l, ϕ) ∈ T , we denote by Hτ the Sobolev space W (R , ϕ(x) dx). Then

D = proj lim Hτ , τ∈T

see [8, Chapter 14, Subsec. 4.3] for details. As the center space H0 = Hτ0 we choose 2 d L (R , dx) (i.e., τ0 = (0, 1)). Thus, we obtain the Gel’fand triple 2 d 0 D ⊂ L (R , dx) ⊂ D . n ∞ d n Note that nuclear space D consists of all functions from C0 ((R ) ) that are d n symmetric in the variables (x1, . . . , xn) ∈ (R ) . d 0 For each x ∈ R , the delta function δx belongs to D , and we will use the notations

D(x) := D(δx),E(x) := E(δx),Q(x) := Q(δx) (the latter operator being deﬁned for a given ﬁxed monic polynomial sequence on D0). (k) 0 k (k) k Below, for any F ∈ D and f ∈ D , k ∈ N0, we denote (k) (k) (k) (k) hF (x1, . . . , xk), f (x1, . . . , xk)i := hF , f i.

15 (n) ∞ 0 (n) Theorem 4.1. Let (P )n=0 be a monic polynomial sequence on D such that P (0) = 0 for all n ∈ N. Let (Q(ζ))ζ∈D0 be the corresponding lowering operators. Then the following conditions are equivalent:

(n) ∞ (BT1) The sequence (P )n=0 is of binomial type. (BT2) For each ζ ∈ D0, Q(ζ) is shift-invariant.

∞ 0 0 k (BT3) There exists a sequence (Bk)k=1 with Bk ∈ L(D , D ), k ≥ 2 and B1 = 1, the identity operator on D0, such that for all ζ ∈ D0 and P ∈ P(D0),

∞ X 1 (Q(ζ)P )(ω) = (B ζ)(x , . . . , x ), (D(x ) ··· D(x )P )(ω) , ω ∈ D0. k! k 1 k 1 k k=1 (4.1)

(n) ∞ (BT4) The monic polynomial sequence (P )n=0 has the generating function

∞ ∞ X 1 X 1 hP (n)(ω), ξ⊗ni = hω⊗n,A(ξ)⊗ni n! n! n=0 n=0 ∞ X 1 n = hω, A(ξ)i = exp hω, A(ξ)i, ω ∈ D0. (4.2) n! n=0 Here ∞ X ⊗k A(ξ) = Akξ ∈ S(D, D), (4.3) k=1 k where Ak ∈ L(D , D), k ≥ 2, and A1 = 1, the identity operator on D, while (4.2) is an equality in S(D, R). P∞ 1 ⊗n ⊗n Remark 4.2. Note that n=0 n! hω ,A(ξ) i ∈ S(D, R) in formula (4.2) is the com- P∞ 1 ⊗n ⊗n position of exp[hω, ξi] := n=0 n! hω , ξ i ∈ S(D, R) and A(ξ) ∈ S(D, D). Remark 4.3. Let A(ξ) ∈ S(D, D) be as in (4.2). Denote

∞ X 1 B(ξ) := B∗ξ⊗k ∈ S(D, D), k! k k=1

where the operators Bk are as in (BT3). It will follow from the proof of Theorem 4.1 that B(ξ) is the compositional inverse of A(ξ), see Deﬁnition A.9, Proposition A.11, and Remark A.12. Remark 4.4. It will also follow from the proof of Theorem 4.1 that, in (BT3), for each ∗ k k ≥ 2, we have Bk = R1,k, the adjoint of the operator R1,k ∈ L(D , D) from formula (3.4).

16 Before proving this theorem, let us ﬁrst note its immediate corollary.

∞ k Corollary 4.5. Consider any sequence (Ak)k=1 with Ak ∈ L(D , D), k ≥ 2, and (n) ∞ 0 A1 = 1. Then there exists a unique sequence (P )n=0 of monic polynomials on D of binomial type that has the generating function (4.2) with A(ξ) given by (4.3).

0 1 (n) ∞ Proof. Define A(ξ) ∈ S(D, D) by formula (4.3). For each ω ∈ D , define ( n! P (ω))n=0 ∈ 0 (n) ∞ F(D ) by formula (4.2). It easily follows from Definitions 3.2 and A.5 that (P )n=0 is a monic polynomial sequence on D0. Furthermore, for n ∈ N, in the representation (3.2) (n) (n) of P (ω), we obtain Un,0 = 0 so that P (0) = 0. Now the statement follows from Theorem 4.1. We will now prove Theorem 4.1. Proof of (BT1) ⇒ (BT2). First, we note that, for any η, ζ ∈ D0, E(ζ)Q(η)1 = Q(η)E(ζ)1 = 0. (4.4)

Next, using the the binomial identity (3.6), we get, for all ξ ∈ D and n ∈ N, n X n Q(η)E(ζ)hP (n), ξ⊗ni = hP (n−k)(ζ), ξ⊗(n−k)iQ(η)hP (k), ξ⊗ki k k=0 n X n = hη, ξi khP (n−k)(ζ), ξ⊗(n−k)ihP (k−1), ξ⊗(k−1)i k k=1 n X n − 1 = nhη, ξi hP (n−k)(ζ), ξ⊗(n−k)ihP (k−1), ξ⊗(k−1)i k − 1 k=1 n−1 X n − 1 = nhη, ξi hP (n−k−1)(ζ), ξ⊗(n−k−1)ihP (k), ξ⊗ki k k=0 = nhη, ξiE(ζ)hP (n−1), ξ⊗(n−1)i = E(ζ)Q(η)hP (n), ξ⊗ni. (4.5) By (4.4), (4.5), and Lemma 3.5, we get E(ζ)Q(η)P = Q(η)E(ζ)P for all P ∈ P(D0).

In order to prove the implication (BT2) ⇒ (BT1), we ﬁrst need the following

(n) ∞ Proposition 4.6 (Polynomial expansion). Let (P )n=0 be a monic polynomial sequence on D0 such that P (n)(0) = 0 for all n ∈ N, or, equivalently, for P (n) being of the form (3.2), Un,0 = 0. Let (Q(ζ))ζ∈D0 be the corresponding lowering operators. Then, for each P ∈ P(D0), we have

∞ X 1 P (ω) = P (k)(ω)(x , . . . , x ), (Q(x ) ··· Q(x )P )(0) , ω ∈ D0. (4.6) k! 1 k 1 k k=0

17 Here, for k = 0, we set Q(x1) ··· Q(xk)P := P .

d Proof. For x1, . . . , xk ∈ R , k ∈ N, ξ ∈ D, and n ∈ N, we have (n) ⊗n (n−k) ⊗(n−k) Q(x1) ··· Q(xk)hP , ξ i = (n)kξ(x1) ··· ξ(xk)hP , ξ i, (4.7)

where (n)k := n(n − 1) ··· (n − k + 1). Note that (n)k = 0 for k > n. Hence, for k, n ∈ N0, one ﬁnds (n) ⊗n ⊗n Q(x1) ··· Q(xk)hP , ξ i (0) = δk,nn! ξ (x1, . . . , xn),

where δk,n denotes the Kronecker symbol. Thus,

∞ X 1 hP (n)(ω), ξ⊗ni = P (k)(ω)(x , . . . , x ), Q(x ) ··· Q(x )hP (n), ξ⊗ni (0) . k! 1 k 1 k k=0 Hence, by Lemma 3.5, formula (4.6) holds for a generic P ∈ P(D0).

Proof of (BT2) ⇒ (BT1). Let ζ ∈ D0, ξ ∈ D, and n ∈ N. An application of Proposi- tion 4.6 to the polynomial P = E(ζ)hP (n), ξ⊗ni yields

∞ X 1 hP (n)(ω + ζ), ξ⊗ni = P (k)(ω)(x , . . . , x ), (Q(x ) ··· Q(x )E(ζ)hP (n), ξ⊗ni)(0) , k! 1 k 1 k k=0 (4.8) and by (BT2) and (4.7), we have, for k ∈ N, (n) ⊗n (n) ⊗n Q(x1) ··· Q(xk)E(ζ)hP , ξ i (0) = E(ζ)Q(x1) ··· Q(xk)hP , ξ i (0) (n) ⊗n = Q(x1) ··· Q(xk)hP , ξ i (ζ) (n−k) ⊗(n−k) = (n)kζ(x1) ··· ζ(xk)hP (ζ), ξ i. Hence,

n X n hP (n)(ω + ζ), ξ⊗ni = hP (k)(ω), ξ⊗kihP (n−k)(ζ), ξ⊗(n−k)i. k k=0 Thus, we have proved the equivalence of (BT1) and (BT2). To continue the proof of Theorem 4.1, we will need the following result.

(n) ∞ Theorem 4.7 (Operator expansion theorem). Let (P )n=0 be a monic polynomial 0 sequence on D of binomial type, and let (Q(ζ))ζ∈D0 be the corresponding lowering operators. A linear operator T acting on P(D0) is continuous and shift-invariant if (k) ∞ 0 0 and only if there is a (G )k=0 ∈ F(D ) such that, for each P ∈ P(D ), ∞ X 1 (TP )(ω) = hG(k)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ω)i, ω ∈ D0. (4.9) k! 1 k 1 k k=0

18 (k) k In the latter case, for each k ∈ N0 and f ∈ D , hG(k), f (k)i = T hP (k), f (k)i (0). (4.10) Remark 4.8. Below we will sometimes write formula (4.9) in the form ∞ X 1 T = G(k)(x , . . . , x ),Q(x ) ··· Q(x ) . k! 1 k 1 k k=0 Proof of Theorem 4.7. Let n ∈ N, ω, ζ ∈ D0, and ξ ∈ D. By (4.8), we have ∞ X 1 E(ζ)hP (n), ξ⊗ni(ω) = P (k)(ω)(x , . . . , x ), (Q(x ) ··· Q(x )hP (n), ξ⊗ni)(ζ) . k! 1 k 1 k k=0 (4.11) Note that formula (4.11) remains true when n = 0. Hence, by Lemma 3.5, for each P ∈ P(D0), ∞ X 1 E(ζ)P (ω) = P (k)(ω)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ζ) . (4.12) k! 1 k 1 k k=0 Assume T ∈ L(P(D0)) is shift-invariant. Swapping ω and ζ in (4.12) and applying T to this equality, we get, for any ω, ζ ∈ D0 and P ∈ P(D0), ∞ X 1 (TE(ω)P )(ζ) = T P (k)(·)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ω) (ζ). (4.13) k! 1 k 1 k k=0 By shift-invariance, the left hand side of (4.13) is equal to (TP )(ω + ζ). In particular, this holds for ζ = 0: ∞ X 1 (TP )(ω) = T P (k)(·)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ω) (0). (4.14) k! 1 k 1 k k=0

(k) 0 k Let G ∈ D , k ∈ N0, be deﬁned by (4.10). Then (4.9) follows from (4.14). (k) ∞ 0 Conversely, let (G )k=0 ∈ F(D ) be ﬁxed, and let T be given by (4.9). As easily seen, T ∈ L(P(D0)). For each P ∈ P(D0) and ζ ∈ D0, we get from (4.9) and (BT2): ∞ X 1 (TE(ζ)P )(ω) = G(k)(x , . . . , x ), (E(ζ)Q(x ) ··· Q(x )P )(ω) k! 1 k 1 k k=0 ∞ X 1 = G(k)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ω + ζ) k! 1 k 1 k k=0 = (TP )(ω + ζ) = (E(ζ)TP )(ω). Therefore, the operator T is shift-invariant. Moreover, it easily follows from (4.9) and (4.7) that (4.10) holds.

19 Note that the statement (BT3) ⇒ (BT2) follows immediately from Theorem 4.7. Proof of (BT2) ⇒ (BT3). Let ζ ∈ D0. We apply Theorem 4.7 to the sequence of monomials and its family of lowering operators, (D(η))η∈D0 , and the shift-invariant operator Q(ζ). By using also formula (3.4), we obtain

∞ X 1 Q(ζ) = G(k)(ζ, x , . . . , x ),D(x ) ··· D(x ) , (4.15) k! 1 k 1 k k=1 where

(k) (k) ⊗k (k) (k) hG (ζ, x1, . . . , xk), f (x1, . . . , xk)i = Q(ζ)h· , f i (0) = hζ, R1,kf i (4.16)

(k) k for all k ∈ N and f ∈ D . Here, we set R1,1 := 1, the identity operator on D. For ∗ 0 0 k k ∈ N, we denote Bk := R1,k ∈ L(D , D ). Note that B1 = 1, the identity operator on D0. By (4.16), (k) G (ζ, ·) = Bkζ, k ≥ 1. (4.17) Formulas (4.15), (4.17) imply (BT3). Thus, we have proved the equivalence of (BT1), (BT2), and (BT3). According to Theorem 4.7, under the conditions assumed therein, there is a one-to- 1 (k) ∞ one correspondence between shift-invariant operators T and sequences ( k! G )k=0 ∈ F(D0). We noted above that the space S(P(D0)) of shift-invariant operators is an algebra under the product of operators, while F(D0) is a commutative algebra under the symmetric tensor product.

(n) ∞ Theorem 4.9 (The isomorphism theorem). Let (P )n=0 be a sequence of monic poly- 0 nomials on D of binomial type, and let (Q(ζ))ζ∈D0 be the corresponding lowering operators. Then, the correspondence given by Theorem 4.7, ∞ 0 1 (k) 0 S(P(D )) 3 T 7→ JT := G ∈ F(D ), k! k=0 is an algebra isomorphism.

Proof. In view of Theorem 4.7, we only have to prove that, for any S,T ∈ S(P(D0)), J(ST ) = JS JT. (4.18)

Let 1 ∞ 1 ∞ JS = F (k) ,JT = G(k) . k! k=0 k! k=0

20 By Theorem 4.7, for all ξ ∈ D and ω ∈ D0,

n X n T hP (n), ξ⊗ni (ω) = hG(k), ξ⊗kihP (n−k)(ω), ξ⊗(n−k)i, (4.19) k k=0 and a similar expression holds for S. Therefore,

ST hP (n), ξ⊗ni (ω) n X n = hG(k), ξ⊗ki ShP (n−k), ξ⊗(n−k)i (ω) k k=0 n n−k X n X n − k = hG(k), ξ⊗ki hF (i), ξ⊗iihP (n−k−i)(ω), ξ⊗(n−k−i)i k i k=0 i=0 n n−k X X n! = hG(k) F (i), ξ⊗(k+i)ihP (n−k−i)(ω), ξ⊗(n−k−i)i k! i!(n − k − i)! k=0 i=0 n * j + X n X j = G(k) F (j−k), ξ⊗j hP (n−j)(ω), ξ⊗(n−j)i. j k j=0 k=0

From here and (4.19), formula (4.18) follows. As an immediate consequence of Theorem 4.9, we conclude

Corollary 4.10. Any two shift-invariant operators commute.

Corollary 4.11. Let the conditions of Theorem 4.9 be satisfied and let the operator J : S(P(D0)) → F(D0) be defined as in that theorem. Define J : S(P(D0)) → S(D, R) by J := IJ. Here I : F(D0) → S(D, R) is defined by (2.11). Then, for each T ∈ S(P(D0)), we have ∞ X 1 (J T )(ξ) = T hP (n), ξ⊗ni (0). (4.20) n! n=0 Furthermore, J is an algebra isomorphism, i.e., for any S,T ∈ S(P(D0)), we have J (ST )(ξ) = (J S)(ξ)(J T )(ξ). (4.21)

Proof. Formula (4.20) follows Theorem 4.7 and the deﬁnition of J . Formula (4.21) is a consequence of (2.12) and Theorem 4.9.

Corollary 4.12. Let T ∈ S(P(D0)). The operator T is invertible if and only if T 1 6= 0. Furthermore, if T 1 6= 0, then T −1 ∈ S(P(D0)).

21 Proof. If T 1 = 0, then the kernel of T is not equal to {0}. Hence, T is not invertible. Assume T 1 6= 0. Let the isomorphism J from Theorem 4.9 be constructed through the monomials and the corresponding lowering operators D(ζ), ζ ∈ D0. So formula (4.20) becomes ∞ X 1 (J T )(ξ) = T h·⊗n, ξ⊗ni (0). (4.22) n! n=0 Since T 1 6= 0, formula (4.22), Corollary 4.11, and Proposition A.1 imply the existence of an operator S ∈ S(P(D0)) such that ST = TS = 1. Hence, the operator T is invertible and T −1 = S ∈ S(P(D0)). Proof of (BT3) ⇒ (BT4). Let the isomorphism J from Theorem 4.9 be constructed through the monomials and the corresponding lowering operators D(ζ), ζ ∈ D0. By (4.22), (J D(ζ))(ξ) = hζ, ξi. Thus, by Lemma 2.20 and the isomorphism theorem,

∞ ∞ X 1 X 1 (J E(ζ))(ξ) = (J D(ζ)n)(ξ) = hζ⊗n, ξ⊗ni = exp[hζ, ξi]. (4.23) n! n! n=0 n=0

Let G(k) ∈ D0 k, k ∈ N. Then formula (4.22) with

(k) T = G (x1, . . . , xk),D(x1) ··· D(xk)

yields (k) (k) ⊗k (J G (x1, . . . , xk),D(x1) ··· D(xk) )(ξ) = hG , ξ i. Therefore, condition (BT3) gives, for each ζ ∈ D0,

∞ ∞ X 1 X 1 (J Q(ζ))(ξ) = hB ζ, ξ⊗ki = hζ, R ξ⊗ki. (4.24) k! k k! 1,k k=1 k=1

∗ In the latter equality we used the fact that R1,k = Bk, see the proof of (BT2) ⇒ (BT3). d Choosing in (4.24) ζ = δx, x ∈ R , we obtain

∞ X 1 (J Q(x))(ξ) = R ξ⊗k (x), k! 1,k k=1

d and, more generally, by Theorem 4.9, for any x1, . . . , xk ∈ R , k ∈ N,

k ∞ ! Y X 1 (J Q(x ) ··· Q(x ))(ξ) = R ξ⊗n (x ) . (4.25) 1 k n! 1,n i i=1 n=1

22 By (4.22) and (4.25), we get, for each G(k) ∈ D0 k, k ∈ N,

(k) J hG (x1, . . . , xk),Q(x1) ··· Q(xk)i (ξ) (k) = G (x1, . . . , xk), J (Q(x1) ··· Q(xk))(ξ) * k ∞ !+ Y X 1 = G(k)(x , . . . , x ), R ξ⊗n (x ) 1 k n! 1,n i i=1 n=1 * ∞ !⊗k+ X 1 = G(k), R ξ⊗n . (4.26) n! 1,n n=1 By (4.12) and (4.26),

∞ * ∞ !⊗k+ X 1 X 1 (J E(ζ))(ξ) = P (k)(ζ), R ξ⊗n . (4.27) k! n! 1,n k=0 n=1 Formulas (4.23) and (4.27) imply

∞ * ∞ !⊗k+ X 1 X 1 P (k)(ζ), R ξ⊗n = exp[hζ, ξi]. (4.28) k! n! 1,n k=0 n=1

P∞ ⊗k By Proposition A.11, we ﬁnd the compositional inverse A(ξ) = k=1 Akξ ∈ P∞ 1 ⊗n S(D, D) of n=1 n! R1,nξ ∈ S(D, D), and by Remark A.12, we have A1 = 1. For- mula (4.2) now follows from (4.28) and Proposition A.7. Remark 4.13. It follows from the proof of Proposition A.11 that, for k ≥ 2, the operators Ak are given by the recurrence formula

k X 1 X Ak = − R1,n Al1 · · · Aln . (4.29) n! n n=2 (l1,...,ln)∈N l1+···+ln=k

Proof of (BT4) ⇒ (BT1). By (BT4), we have, for any ω, ζ ∈ D0,

∞ ∞ ! ∞ ! X 1 X 1 X 1 hP (n)(ω + ζ), ξ⊗ni = hP (n)(ω), ξ⊗ni hP (k)(ζ), ξ⊗ki n! n! k! n=0 n=0 k=0 ∞ * n + X 1 X n = P (k)(ω) P (n−k)(ζ), ξ⊗n , n! k n=0 k=0 which implies (BT1). This concludes the proof of Theorem 4.1.

23 0 Deﬁnition 4.14. Let (Q(ζ))ζ∈D0 be a family of operators from L(P(D )). We say that (Q(ζ))ζ∈D0 is a family of delta operators if the following conditions are satisﬁed: (i) Each Q(ζ) is shift-invariant; (ii) For each ζ ∈ D0 and each ξ ∈ D, Q(ζ)h·, ξi = hζ, ξi; (4.30)

(iii) Q(ζ) linearly depends on ζ ∈ D0. Furthermore, for each k ≥ 2, the mapping 0 0 k D 3 ζ 7→ Bkζ ∈ D deﬁned by (k) ⊗k (k) (k) k hBkζ, f i := Q(ζ)h· , f i (0), f ∈ D , (4.31) belongs to L(D0, D0 k). It is a straightforward consequence of Theorem 4.1 that, for any monic polynomial (n) ∞ sequence (P )n=0 of binomial type, the corresponding family (Q(ζ))ζ∈D0 of lowering operators is a family of delta operators.

Proposition 4.15. Let (Q(ζ))ζ∈D0 be a family of delta operators. Then, there exists a (n) ∞ unique monic polynomial sequence (P )n=0 of binomial type for which (Q(ζ))ζ∈D0 is the family of lowering operators. Proof. Let ζ, ω ∈ D0 and ξ ∈ D . By shift-invariance of Q(ζ) and (4.30), we get hζ, ξi = E(ω)hζ, ξi = E(ω)Q(ζ)h·, ξi = Q(ζ)E(ω)h·, ξi = Q(ζ)h· + ω, ξi = Q(ζ)h·, ξi + hω, ξiQ(ζ)1 = hζ, ξi + hω, ξiQ(ζ)1, which implies that Q(ζ)1 = 0. Hence, by Theorem 4.7,(4.30), and (4.31), we have ∞ X 1 Q(ζ) = (B ζ)(x , . . . , x ),D(x ) ··· D(x )i, k! k 1 k 1 k k=1 0 with B1 := 1, the identity operator on D . Let A1 := 1 be the identity operator on D, and for k ≥ 2, let the operators Ak ∈ k ∗ L(D , D) be deﬁned by the recurrence formula (4.29) with R1,k := Bk. Thus, A(ξ) = P∞ ⊗k P∞ 1 ∗ ⊗n k=1 Akξ ∈ S(D, D) is the compositional inverse of n=1 n! Bnξ ∈ S(D, D) (n) ∞ 0 Let (P )n=0 be the monic polynomial sequence on D of binomial type that has the generating function (4.2) with A(ξ) given by (4.3), see Corollary 4.5. By Remark 4.3, (n) ∞ (P )n=0 is the unique required polynomial sequence.

Deﬁnition 4.16. Let (Q(ζ))ζ∈D0 be a family of delta operators. The corresponding (n) ∞ monic polynomial sequence (P )n=0 of binomial type given by Proposition 4.15 is called the basic sequence for (Q(ζ))ζ∈D0 .

Proposition 4.17. (Q(ζ))ζ∈D0 is a family of delta operators if and only if there exists ∞ 0 0 k a sequence (Bk)k=1, with Bk ∈ L(D , D ), such that B1 = 1 and (4.1) holds. Proof. The statement follows immediately from Theorem 4.1 and Proposition 4.15.

24 5 Lifting of polynomials on R of binomial type

∞ Let (pn)n=0 be a monic polynomial sequence on R of binomial type, and let Q be its delta operator, that is, Qpn = npn−1 for each n ∈ N0. According to the one-dimensional (classical) version of Theorem 4.1 (see e.g. [24]), Q has a formal expansion

∞ X bk Q = Dk = q(D), (5.1) k! k=1

where (bk)k∈N is a sequence of real numbers such that b1 = 1, D is the diﬀerentiation operator and ∞ X bk q(t) := tk (5.2) k! k=1 is a formal power series in t ∈ R. Furthermore, ∞ X un p (t) = exp[ta(u)], (5.3) n! n n=0 where the formal power series in u ∈ R, ∞ X k a(u) = aku (5.4) k=1

is the compositional inverse of q. In particular, a1 = 1. We will now lift the sequence ∞ 0 of polynomials (pn)n=0 to a monic polynomial sequence on D of binomial type. k For each k ∈ N, we define an operator Dk ∈ L(D , D) by (k) (k) (k) k d (Dkf )(x) := f (x, . . . , x), f ∈ D , x ∈ R (5.5) ∗ 0 0 k (D1 being the identity operator on D). The adjoint operator Dk ∈ L(D , D ) satisfies ∗ (k) (k) 0 (k) k hDkζ, f i = hζ(x), f (x, . . . , x)i, ζ ∈ D , f ∈ D . In particular, ∗ ⊗k k 0 hDkζ, ξ i = hζ, ξ i, ζ ∈ D , ξ ∈ D. (5.6) 0 0 k We now define an operator Bk : D → D by

∗ Bk := bkDk, k ∈ N, (5.7)

where the numbers bk are as in (5.1). Let (Q(ζ))ζ∈D0 be the family of delta operators given by (4.1), see Proposition 4.17. By (5.6) and (5.7), we then have

∞ X bk Q(x) = Dk(x) = q(D(x)), x ∈ d, (5.8) k! R k=1

25 and moreover, ∞ X bk Q(ζ) = ζ(x), Dk(x) = ζ(x), q(D(x)) . (5.9) k! k=1 (n) ∞ Let (P )n=0 be the basic sequence for (Q(ζ))ζ∈D0 . Thus, in view of (5.1) and (5.8), (n) ∞ ∞ we may think of (P )n=0 as the lifting of the monic polynomial sequence (pn)n=0 of binomial type. (n) ∞ As easily seen, the generating function of (P )n=0 is given by (4.2) with the oper- k ators Ak ∈ L(D , D) given by

Ak = akDk, k ∈ N, where the numbers ak are as in (5.4). Therefore, by (4.3), ∞ X k A(ξ) = akξ = a(ξ). (5.10) k=1 Hence, ∞ "* ∞ +# X 1 X hP (n)(ω), ξ⊗ni = exp ω(x), a ξk(x) = exp hω, a(ξ)i. (5.11) n! k n=0 k=1 Thus, this generating function can be though of as the lifting of the generating function (5.3). Recall that a set partition π of a set X 6= ∅ is an (unordered) collection of disjoint nonempty subsets of X whose union equals X . We denote by P(n) the collection of all set partitions of X = {1, 2, . . . , n}. For a set B ⊂ {1, . . . , n}, we denote by |B| the cardinality of B. (n) ∞ 0 Proposition 5.1. Let (P )n=0 be the monic polynomial sequence on D of binomial type that has the generating function (5.11). For k ∈ N, denote αk := akk!, so that ∞ X αk a(u) = uk. (5.12) k! k=1 Then, for any n ∈ N, ω ∈ D0, and ξ ∈ D, (n) ⊗n X Y |B| hP (ω), ξ i = α|B|hω, ξ i, (5.13) π∈P(n) B∈π or equivalently

n (n) X X P (ω) = α|B1|α|B1| ··· α|Bk|

k=1 {B1,B2,...,Bk}∈P(n) × ∗ ω⊗|B1| ∗ ω⊗|B2| · · · ∗ ω⊗|Bk|. (5.14) D|B1| D|B2| D|Bk|

26 Proof. It follows immediately from the form of the generating function (5.11) that

n X n! X hP (n)(ω), ξ⊗ni = a ··· a hω, ξi1 i · · · hω, ξik i k! i1 ik k k=1 (i1,...,ik)∈N i1+···+ik=n X n! = αj1 αj2 ··· αjn j1 j2 jn 1 2 n n j1!j2! ··· jn! (1!) (2!) ··· (n!) (j1,j2...,jn)∈N0 j1+2j2+···+njn=n × hω, ξij1 hω, ξ2ij2 · · · hω, ξnijn . which implies (5.13), hence also (5.14).

We denote by M(Rd) the space of all signed Radon measures on Rd, i.e., the set of d d d all signed measures η on (R , B(R )) such that |η|(Λ) < ∞ for all Λ ∈ B0(R ). Here d d d B(R ) denotes the Borel σ-algebra on R , B0(R ) denotes the collection of all bounded sets Λ ∈ B(Rd), and |η| denotes the variation of η ∈ M(Rd). d n d n Further, let Bsym((R ) ) denote the sub-σ-algebra of B((R ) ) that consists of all symmetric sets ∆ ∈ B((Rd)n), i.e., for each permutation σ ∈ S(n), ∆ is an invariant set for the mapping

d n d n (R ) 3 (x1, . . . , xn) 7→ (xσ(1), . . . , xσ(n)) ∈ (R ) .

d n d n d n We denote by Msym((R ) ) the space of all signed Radon measures on ((R ) , Bsym((R ) )).

(n) ∞ 0 Corollary 5.2. Let (P )n=0 be the monic polynomial sequence on D of binomial type that has the generating function (5.11). Then, for each η ∈ M(Rd) and n ∈ N, we have (n) d n d P (η) ∈ Msym((R ) ). Furthermore, for each Λ ∈ B0(R ),

(n) n P (η) (Λ ) = pn(η(Λ)), n ∈ N, (5.15)

∞ where (pn)n=0 is the polynomial sequence on R with generating function (5.3).

d ∗ d j (j) j Proof. Let η ∈ M(R ). For each j ∈ N, Dj η ∈ Msym((R ) ), since for each f ∈ D Z ∗ (j) (j) hDj η, f i = f (x, . . . , x) dη(x). Rd ∗ Note that the measure Dj η is concentrated on the set

d j {(x1, x2, . . . , xj) ∈ (R ) | x1 = x2 = ··· = xj}.

(n) d n By formula (5.14), we therefore get P (η) ∈ Msym((R ) ).

27 d Fix any Λ ∈ B0(R ). Set ξ := uχΛ, where u ∈ R and χΛ denotes the indicator function of Λ. It easily follows from (5.11) by an approximation argument that ∞ ∞ X un X 1 P (n)(η)(Λn) = hP (n)(η), ξ⊗ni n! n! n=0 n=0 Z = exp a(ξ(x)) dη(x) Rd = exp [η(Λ)a(u)] . (5.16) In formula (5.16), hP (n)(η), ξ⊗ni denotes the integral of the function ξ⊗n with respect to the measure P (n)(η). Formula (5.15) now follows from (5.3) and (5.16).

(n) ∞ The following proposition shows that the lifted polynomials (P )n=0 have an ad- ditional property of binomial type. (n) ∞ 0 Proposition 5.3. Let (P )n=0 be the monic polynomial sequence on D of binomial type that has the generating function (5.11). Let ξ, φ ∈ D be such that d d {x ∈ R | ξ(x) 6= 0} ∩ {x ∈ R | φ(x) 6= 0} = ∅. (5.17) Then, for any ω ∈ D0 and k, n ∈ N, hP (k+n)(ω), ξ⊗k φ⊗ni = hP (k)(ω), ξ⊗kihP (n)(ω), φ⊗ni. (5.18) Therefore, for each n ∈ N, n X n hP (n)(ω), (ξ + φ)⊗ni = hP (k)(ω), ξ⊗kihP (n−k)(ω), φ⊗(n−k)i. (5.19) k k=0 Proof. We have ∞ ∞ n X 1 X X 1 hP (n)(ω), (ξ + φ)⊗ni = hP (n)(ω), ξ⊗k φ⊗(n−k)i n! k!(n − k)! n=0 n=0 k=0 ∞ ∞ X X 1 = hP (n+k)(ω), φ⊗n ξ⊗ki, (5.20) k! n! k=0 n=0 and by (5.11) and (5.17) ∞ X 1 hP (n)(ω), (ξ + φ)⊗ni = exp hω, a(ξ + φ)i n! n=0 = exp hω, a(ξ)i exp hω, a(φ)i ∞ ∞ X X 1 = hP (k)(ω), ξ⊗kihP (n)(ω), φ⊗ni. (5.21) k! n! k=0 n=0 Formulas (5.20), (5.21) imply (5.18), hence also (5.19). We will now consider examples of sequences of lifted polynomials of binomial type.

28 5.1 Falling factorials on D0

∞ The classical falling factorials is the sequence (pn)n=0 of monic polynomials on R of binomial type that are explicitly given by

pn(t) = (t)n := t(t − 1)(t − 2) ··· (t − n + 1).

The corresponding delta operator is Q = eD − 1, so that Q is the diﬀerence operator (Qp)(t) = p(t + 1) − p(t). Here p belongs to P(R), the space of polynomials on R. The generating function of the falling factorials is

∞ X un (t) = exp[t log(1 + u)] = (1 + u)t. n! n n=0 One also deﬁnes an extension of the binomial coeﬃcient, t 1 t(t − 1)(t − 2) ··· (t − n + 1) := (t) = , (5.22) n n! n n!

which becomes the classical binomial coeﬃcient for t ∈ N, t ≥ n. (n) ∞ Let us now consider the corresponding lifted sequence of polynomials, (P )n=0. We will call these polynomials the falling factorials on D0. By analogy with the one- (n) 0 dimensional case, we will write (ω)n := P (ω) for ω ∈ D . By (5.8), Q(x) = eD(x) − 1. Hence, by Boole’s formula,

d 0 (Q(x)P )(ω) = P (ω + δx) − P (ω), x ∈ R ,P ∈ P(D ), (5.23) and by (5.9),

0 0 (Q(ζ)P )(ω) = ζ(x),P (ω + δx) − P (ω) , ζ ∈ D ,P ∈ P(D ).

Further, by (5.11), the generating function is given by

∞ X 1 h(ω) , ξ⊗ni = exp hω, log(1 + ξ)i. (5.24) n! n n=0 Proposition 5.4. The falling factorials on D0 have the following explicit form:

(ω)0 = 1;

(ω)1 = ω;

(ω)n(x1, . . . , xn) = ω(x1)(ω(x2) − δx1 (x2))

× (ω(x3) − δx1 (x3) − δx2 (x3)) ··· (ω(xn) − δx1 (xn) − · · · − δxn−1 (xn)) (5.25) for n ≥ 2.

29 ∞ 0 Proof. Let ((ω)n)n=0 denote the monic polynomial sequence on D deﬁned by formula (5.25). Note that, for n ∈ N,(ω)n = 0 for ω = 0. It can be easily shown by induction that the polynomials (ω)n satisfy the following recurrence relation:

(ω)0 = 1; ⊗(n+1) ⊗(n+1) 2 ⊗(n−1) h(ω)n+1, ξ i = h(ω)n ω, ξ i − nh(ω)n, ξ ξ i, ξ ∈ D, n ∈ N0. (5.26)

∞ Furthermore, this recurrence relation uniquely determines the polynomials ((ω)n)n=0. It suﬃces to prove that, for each ω ∈ D0, n ∈ N, ξ ∈ D, and x ∈ Rd,

⊗n ⊗(n−1) Q(x)h(·)n, ξ i (ω) = nξ(x)h(ω)n−1, ξ i,

or equivalently, by (5.23),

⊗n ⊗n ⊗(n−1) h(ω + δx)n, ξ i = h(ω)n, ξ i + nξ(x)h(ω)n−1, ξ i. (5.27)

We prove this formula by induction. It trivially holds for n = 1. Assume that (5.27) holds for 1, 2, . . . , n and let us prove it for n + 1. Using our assumption, the recurrence relation (5.26) and the polarization identity, we get

⊗(n+1) ⊗n 2 ⊗(n−1) h(ω + δx)n+1, ξ i = h(ω + δx)n, ξ ihω + δx, ξi − nh(ω + δx)n, ξ ξ i ⊗n ⊗(n−1) = h(ω)n, ξ i + nξ(x)h(ω)n−1, ξ i hω, ξi + ξ(x) 2 ⊗(n−1) 2 ⊗(n−1) − n h(ω)n, ξ ξ i + ξ (x)h(ω)n−1, ξ i 2 ⊗(n−2) + (n − 1)ξ(x)h(ω)n−1, ξ ξ i ⊗(n+1) 2 ⊗(n−1) ⊗n = h(ω)n ω, ξ i − nh(ω)n, ξ ξ i + ξ(x)h(ω)n, ξ i ⊗n 2 ⊗(n−2) + nξ(x) h(ω)n−1 ω, ξ i − (n − 1)h(ω)n−1, ξ ξ i ⊗(n+1) ⊗n = h(ω)n+1, ξ i + nξ(x)h(ω)n, ξ i.

Remark 5.5. Note that the recurrence relation (5.26) satisfied by the polynomials on D0 with generating function (5.24) was already discussed in [7]. 0 Since we have interpreted (ω)n as a falling factorial on D , we naturally define ‘ω ω 1 choose n’ by n := n! (ω)n, compare with (5.22). We denote by Γ the configuration space over Rd, i.e., the space of all Radon measures d P∞ γ ∈ M(R ) that are of the form γ = i=1 δxi , where xi 6= xj if i 6= j. (We can P∞ obviously identify the configuration γ = i=1 δxi with the (locally finite) set {xi}i∈N.) The following result is immediate. P∞ Corollary 5.6. For each γ = i=1 δxi , formula (1.1) holds.

30 γ Remark 5.7. Polynomials n play a crucial role in the theory of point process (i.e., Γ-valued random variables), see e.g. [18]. More precisely, given a probability space (Ξ, F, P) and a point process γ :Ξ → Γ, the nth correlation measure of γ is deﬁned as (n) d n d n the (unique) measure σ on (R ) , Bsym((R ) ) that satisﬁes Z γ (n) (n) (n) (n) n (n) E , f = f (x1, . . . , xn) dσ (x1, . . . , xn) for all f ∈ D , f ≥ 0. n (Rd)n

Here E denotes the expectation with respect to the probability measure P. Under very (n) ∞ mild conditions on the point process γ, the correlation measures (σ )n=1 uniquely identify the distribution of γ on Γ. In the case where each measure σ(n) is absolutely continuous with respect to the Lebesgue measure, one deﬁnes the nth correlation func- (n) tion of the point process γ, denote by k (x1, . . . , xn), as follows: 1 dσ(n)(x , . . . , x ) = k(n)(x , . . . , x ) dx ··· dx , 1 n n! 1 n 1 n or equivalently Z (n) (n) (n) E (γ)n, f = f (x1, . . . , xn)k (x1, . . . , xn) dx1 ··· dxn. (Rd)n

d Corollary 5.8. For each Λ ∈ B0(R ), γ ∈ Γ and n ∈ N, we have, γ γ(Λ) (Λn) = . n n

Proof. The result is obvious by Corollary 5.6, or alternatively, by Corollary 5.2. Remark 5.9. In view of Corollaries 5.6 and 5.8, for the falling factorials on D0, the set Γ plays a role similar to that played by the set N for the falling factorials on R.

5.2 Rising factorials on D0

∞ The classical rising factorials is the sequence (pn)n=0 of monic polynomials on R of binomial type that are explicitly given by

n pn(t) = (t) := t(t + 1)(t + 2) ··· (t + n − 1).

The corresponding delta operator is Q = 1 − e−D, so that Q is the diﬀerence operator (Qp)(t) = p(t) − p(t − 1) for p ∈ P(R). The generating function of the rising factorials is given by ∞ X un (t)n = exp[−t log(1 − u)] = (1 − u)−t. n! n=0

31 One also has the following connection between the rising factorials and the falling n n factorials: (t) = (−1) (−t)n. (n) ∞ Let us now consider the corresponding lifted sequence of polynomials, (P )n=0. We will call these polynomials the rising factorials on D0, and we will write (ω)n := P (n)(ω) for ω ∈ D0. By (5.8), Q(x) = 1 − e−D(x). Hence, by Boole’s formula,

d 0 (Q(x)P )(ω) = P (ω) − P (ω − δx), x ∈ R ,P ∈ P(D ), and by (5.9),

0 0 (Q(ζ)P )(ω) = ζ(x),P (ω) − P (ω − δx) , ζ ∈ D ,P ∈ P(D ). By (5.11), the generating function is equal to

∞ X 1 h(ω)n, ξ⊗ni = exp hω, − log(1 − ξ)i. (5.28) n! n=0 n n Proposition 5.10. We have (ω) = (−1) (−ω)n for all n ∈ N, and the following explicit formulas hold: (ω)0 = 1; (ω)1 = ω; n (ω) (x1, . . . , xn) = ω(x1)(ω(x2) + δx1 (x2))

× (ω(x3) + δx1 (x3) + δx2 (x3)) ··· (ω(xn) + δx1 (xn) + ··· + δxn−1 (xn)) for n ≥ 2. Proof. By (5.24),

∞ ∞ X 1 X 1 h(−1)n(−ω) , ξ⊗ni = h(−ω) , (−ξ)⊗ni = exp hω, − log(1 − ξ)i. n! n n! n n=0 n=0 n n 0 Hence, by (5.28), (ω) = (−1) (−ω)n for all ω ∈ D and n ∈ N. From this and Proposition 5.4, the statement follows.

5.3 Abel polynomials on D0

Let us ﬁx a parameter α ∈ R\{0}. The classical Abel polynomials on R corresponding ∞ to the parameter α is the monic polynomial sequence (pn)n=0 of binomial type that has the delta operator Q = DeαD, i.e., (Qp)(t) = p0(t + α) for p ∈ P(R). Thus, Q = q(D), where q(u) = ueαu. The generating function of the Abel polynomials is given by

∞ X un p (t) = exp tα−1W (αu), n! n n=0

32 where W is the inverse function of u 7→ ueu (around 0), the so-called Lambert W - function. (n) ∞ Consider the corresponding lifted sequence of polynomials (P )n=0, the Abel polynomials on D0. By Lemma 2.20 and (5.8),

αD(x) d Q(x) = D(x)e = D(x)E(αδx), x ∈ R , and by (5.9),

0 0 Q(ζ)P = ζ(x),D(x)P (· + αδx) , ζ ∈ D ,P ∈ P(D ).

By (5.11), the generating function is equal to

∞ X 1 hP (n)(ω), ξ⊗ni = exp ω, α−1W (αξ) . n! n=0 We have ∞ X (−αk)k−1 α−1W (αu) = uk. k! k=1 Hence, by Proposition 5.1, X Y hP (n)(ω), ξ⊗ni = (−α|B|)|B|−1hω, ξ|B|i. π∈P(n) B∈π

5.4 Laguerre polynomials on D0 of binomial type

Let us recall that the (monic) Laguerre polynomials on R corresponding to a parameter [k] ∞ k ≥ −1, (pn )n=0, have the generating function

∞ X un tu p[k](t) = exp (1 + u)−(k+1). (5.29) n! n 1 + u n=0

∞ In particular, for the parameter k = −1, the Laguerre polynomial sequence (pn)n=0 := [−1] ∞ (pn )n=0 has the generating function

∞ X un tu p (t) = exp . (5.30) n! n 1 + u n=0

∞ Hence, the polynomial sequence (pn)n=0 is of binomial type and its delta operator is Q = q(D) with ∞ u X q(u) = = uk. 1 − u k=1

33 (n) ∞ The corresponding lifted sequence (P )n=0 will be called the Laguerre polynomial sequence on D0 of binomial type. Thus, for each x ∈ Rd, the delta operator Q(x) of (n) ∞ (P )n=0 is given by

∞ D(x) X Q(x) = q(D(x)) = = D(x)k, 1 − D(x) k=1 and the corresponding generating function is

∞ X 1 ξ hP (n)(ω), ξ⊗ni = exp ω, . (5.31) n! 1 + ξ n=0

By Proposition 5.1, the polynomial hP (n)(ω), ξ⊗ni has representation (5.13) with k+1 αk = (−1) k! . In view of the factor k! in αk, we can also give the following combinatorial formula for hP (n)(ω), ξ⊗ni. Let W(n) denote the collection of all sets β = {b1, b2, . . . , bk} such that each li bi = (j1, . . . , jli ) is an element of {1, 2, . . . , n} with ju 6= jv if u 6= v, and each j ∈ {1, 2, . . . , n} is a coordinate of exactly one bi ∈ β. For β ∈ W(n) and bi =

(j1, . . . , jli ) ∈ β, we denote |bi| := li. By (5.13), we now get, for the Laguerre polynomials, X Y hP (n)(ω), ξ⊗ni = h−ω, (−ξ)|b|i. (5.32) β∈W(n) b∈β

6 Sheﬀer sequences

Deﬁnition 6.1. Let (Q(ζ))ζ∈D0 be a family of delta operators. We say that a monic 0 (n) ∞ polynomial sequence on D ,(S )n=0 , is a Sheﬀer sequence for the family of delta 0 (n) n operators (Q(ζ))ζ∈D0 if for each ζ ∈ D and f ∈ D , n ∈ N,

Q(ζ)hS(n), f (n)i = hS(n−1), A(ζ)f (n)i, (6.1)

where A(ζ) is the annihilation operator, see Deﬁnition 2.17. Of course, any basic sequence for a family of delta operators is a Sheﬀer sequence for that family of delta operators.

(n) ∞ Theorem 6.2. Let (Q(ζ))ζ∈D0 be a family of delta operators and let (P )n=0 be its (n) ∞ basic sequence, which has the generating function (4.2). Let (S )n=0 be a monic polynomial sequence on D0. Then the following conditions are equivalent:

(n) ∞ (SS1)( S )n=0 is a Sheﬀer sequence for the family (Q(ζ))ζ∈D0 .

34 0 (n) n (SS2) There is a unique operator T ∈ S(P(D )) such that, for each f ∈ D , n ∈ N0,

T hS(n), f (n)i = hP (n), f (n)i. (6.2)

(n) ∞ (SS3) The sequence (S )n=0 has the generating function

∞ X 1 exp[hω, A(ξ)i] hS(n)(ω), ξ⊗ni = , ω ∈ D0, ξ ∈ D, (6.3) n! τ(A(ξ)) n=0

where A(ξ) ∈ S(D, D) is given by (4.3) and τ ∈ S(D, R) is such that τ(0) = 1.

(SS4) For each n ∈ N and ω, ζ ∈ D0,

n X n S(n)(ω + ζ) = S(k)(ω) P (n−k)(ζ). (6.4) k k=0

(n) ∞ 0 (0) (SS5) There is a (ρ )n=0 ∈ F(D ) with ρ = 1 such that, for each n ∈ N,

n X n S(n)(ω) = ρ(k) P (n−k)(ω), ω ∈ D0. k k=0

Remark 6.3. For the meaning of the right hand side of formula (6.3), see Proposition A.1 and Remark A.2. Proof of Theorem 6.2. (SS1)⇒(SS2). We deﬁne a linear operator T : P(D0) → P(D0) (n) ∞ (n) ∞ 0 by formula (6.2). Since (P )n=0 and (S )n=0 are monic polynomial sequences on D , we have T ∈ L(P(D0)), see formulas (3.3) and (3.4). Thus, we only have to prove that (k) 0 k T is shift-invariant. For this purpose, ﬁx any G ∈ D , ξ ∈ D, and k ∈ N0. By (6.1) and (6.2), we obtain

(k) (n) ⊗n (k) ⊗k (n−k) ⊗(n−k) T G (x1, . . . , xk),Q(x1) ··· Q(xk)hS , ξ i = (n)khG , ξ iT hS , ξ i (k) ⊗k (n−k) ⊗(n−k) = (n)khG , ξ ihP , ξ i (k) (n) ⊗n = G (x1, . . . , xk),Q(x1) ··· Q(xk)hP , ξ i (k) (n) ⊗n = G (x1, . . . , xk),Q(x1) ··· Q(xk)T hS , ξ i .

Therefore,

(k) (k) T hG (x1, . . . , xk),Q(x1) ··· Q(xk)i = hG (x1, . . . , xk),Q(x1) ··· Q(xk)iT.

Hence, by Theorem 4.7, T is shift-invariant.

35 (SS2)⇒(SS1). By (6.2) with n = 0, we have T 1 = 1. Hence, by Corollary 4.12, the operator T is invertible and T −1 ∈ S(P(D0)). By Corollary 4.10, T −1 commutes with each Q(ζ), ζ ∈ D0. Therefore, for each n ∈ N and ξ ∈ D, we get Q(ζ)hS(n), ξ⊗ni = Q(ζ)T −1hP (n), ξ⊗ni = T −1Q(ζ)hP (n), ξ⊗ni = T −1(nhζ, ξihP (n−1), ξ⊗(n−1)i) = nhζ, ξihS(n−1), ξ⊗(n−1)i. (SS2)⇒(SS3). For a ﬁxed ζ ∈ D0, we apply Theorem 4.7 to the shift-invariant operator E(ζ)T −1. This gives

∞ X 1 (E(ζ)T −1P )(ω) = G(k)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ω) ,P ∈ P(D0), k! 1 k 1 k k=0 where hG(k), f (k)i = E(ζ)T −1hP (k), f (k)i (0) = E(ζ)hS(k), f (k)i (0) = hS(k)(ζ), f (k)i. Thus,

∞ X 1 (E(ζ)T −1P )(ω) = S(k)(ζ)(x , . . . , x ), (Q(x ) ··· Q(x )P )(ω) ,P ∈ P(D0). k! 1 k 1 k k=0 (6.5) For the family of delta operators (D(ζ))ζ∈D0 and its basic sequence (monomials), consider the isomorphism J deﬁned in Corollary 4.11. Hence, by (4.26) and (6.5),

∞ * ∞ !⊗k+ X 1 X 1 (J (E(ζ)T −1))(ξ) = S(k)(ζ), R ξ⊗n . (6.6) k! n! 1,n k=0 n=1 Furthermore, by (4.22),

∞ X 1 (J T )(ξ) = 1 + hτ (k), ξ⊗ki =: τ(ξ), (6.7) k! k=1 where (k) ⊗k ⊗k ⊗k hτ , ξ i := T h· , ξ i (0), ξ ∈ D, k ∈ N. (6.8) (Note that the ﬁrst term in (6.7) is indeed equal to 1, because T maps 1 into 1.) By (4.23) and the isomorphism theorem, exp[hζ, ξi] = (J E(ζ))(ξ) = J (E(ζ)T −1)(ξ)(J T )(ξ). (6.9) By Proposition A.1 (see also Remark A.2) and (6.6)–(6.9), we get

∞ * ∞ !⊗k+ X 1 X 1 exp[hζ, ξi] S(k)(ζ), R ξ⊗n = , (6.10) k! n! 1,n τ(ξ) k=0 n=1

36 k compare with (4.28). We deﬁne operators Ak ∈ L(D , D) by formula (4.29) for k ≥ 2 P∞ ⊗k and A1 := 1. Thus, A(ξ) = k=1 Akξ ∈ S(D, D) is the compositional inverse of P∞ 1 ⊗n n=1 n! R1,nξ ∈ S(D, D). Now formula (6.10) implies (SS3). (SS3)⇒(SS2). For the family of delta operators (D(ζ))ζ∈D0 and its basic sequence (monomials), we construct the isomorphism J . We deﬁne an operator T ∈ S(P(D0)) by T := J −1τ. Since τ (0) = 1, by Proposition A.1 and Corollary 4.11, there exists an operator S ∈ S(P(D0)) satisfying ST = TS = 1. Hence, the operator T is invertible and T −1 = S. Since T −11 = 1/τ (0) = 1, for each f (n) ∈ D n, we obtain

n−1 X T −1h·⊗n, f (n)i(ω) = hω⊗n, f (n)i + hω⊗i, g(i)i (6.11) i=0 for some g(i) ∈ D i, i = 0, 1, . . . , n − 1. Consider the linear operator

n (n) (n) (n) −1 (n) (n) (n) 0 D 3 f 7→ S˜ f := T hP , f i ∈ P (D ). (6.12) Since T −1 ∈ L(P(D0)) and satisﬁes (6.11), and the linear operator

D n 3 f (n) 7→ hP (n), f (n)i ∈ P(n)(D0) is continuous (see Lemma 3.1), we conclude that S˜(n) ∈ L(D n, P(n)(D0)). Hence, by ˜(n) ∞ Lemma 3.1 and (6.11), there exists a monic polynomial sequence (S )n=0 that satisﬁes

(n) (n) (n) (n) (S˜ f )(ω) = hS˜ (ω), f i. (6.13) By (6.12) and (6.13), we obtain

hP (n), f (n)i = T hS˜(n), f (n)i.

It follows from the proof of the implication (SS2)⇒(SS3) that the monic polynomial ˜(n) ∞ (n) ∞ sequence (S )n=0 has the same generating function as (S )n=0, so they coincide. But this implies (6.2). Thus, we have proved that the conditions (SS1), (SS2), and (SS3) are equivalent. (SS2)⇒(SS4). For ω, ζ ∈ D0, ξ ∈ D, and n ∈ N we have hS(n)(ω + ζ), ζ⊗ni = E(ζ)hS(n), ξ⊗ni(ω) = E(ζ)T −1hP (n), ξ⊗ni(ω) = T −1E(ζ)hP (n), ξ⊗ni(ω) = T −1hP (n)(· + ζ), ξ⊗ni(ω)

37 n X n = T −1hP (k), ξ⊗ki)(ω)hP (n−k)(ζ), ξ⊗(n−k)i k k=0 n X n = hS(k)(ω), ξ⊗kihP (n−k)(ζ), ξ⊗(n−k)i. k k=0 (SS4)⇒(SS5). In formula (6.4), swap ω and ζ, set ζ = 0, and denote ρ(n) := S(n)(0). Note that ρ(0) = S(0)(0) = 1. (SS5)⇒(SS1). For each ζ ∈ D0, ξ ∈ D, and n ∈ N, we get from (SS5) n X n Q(ζ)hS(n), ξ⊗ni = hρ(k), ξ⊗ki Q(ζ)hP (n−k), ξ⊗(n−k)i k k=0 n−1 X n = hρ(k), ξ⊗ki(n − k)hζ, ξihP (n−k−1), ξ⊗(n−k−1)i k k=0 n−1 X n − 1 = nhζ, ξi hρ(k), ξ⊗kihP (n−k−1), ξ⊗(n−k−1)i k k=0 = nhζ, ξihS(n−1), ξ⊗(n−1)i.

(n) ∞ Corollary 6.4. Let the conditions of Theorem 6.2 be satisﬁed. Then (S )n=0 is a (n) ∞ Sheﬀer sequence for the family (Q(ζ))ζ∈D0 if and only if the sequence (S )n=0 has the generating function

∞ X 1 hS(n)(ω), ξ⊗ni = exp hω, A(ξ)i − C(A(ξ)), ω ∈ D0, ξ ∈ D, (6.14) n! n=0

P∞ ⊗k P∞ (k) ⊗k where A(ξ) := k=1 Akξ ∈ S(D, D) is as in (4.2) and C(ξ) := k=1hC , ξ i ∈ S(D, R) is such that C(0) = 0. Proof. We note that hω, A(ξ)i ∈ S(D, R) is the composition of hω, ξi ∈ S(D, R) and A(ξ) ∈ S(D, D). As easily seen, exp[hω, A(ξ)i] ∈ S(D, R) in formula (6.3) can be P∞ tn understood as the composition of exp t = n=0 n! ∈ S(R, R) and hω, A(ξ)i ∈ S(D, R), see Deﬁnition A.3. P∞ n+1 tn Consider log(1 + t) = n=1(−1) n ∈ S(R, R). Deﬁne C(ξ) ∈ S(D, R) as the composition of log(1 + t) ∈ S(R, R) and τ(ξ) − 1 ∈ S(D, R) (note that τ(0) − 1 = 0). Then C(0) = 0. By Remark A.8, we obtain τ(ξ) = exp(C(ξ)), the equality in S(D, R). Hence, using again Remark A.8, we conclude that formula (6.3) can be written as (6.14).

(n) ∞ Remark 6.5. According to Theorem 6.2 and Corollary 6.4, any Sheﬀer sequence (S )n=0 is completely identiﬁed by a family of delta operators, (Q(ζ))ζ∈D0 , and a formal series C ∈ S(D, R) with C(0) = 0.

38 (n) ∞ Corollary 6.6. Under the conditions of Theorem 6.2, assume that (S )n=0 is a Shef- (n) ∞ 0 (0) fer sequence. Let (κ )n=0 ∈ F(D ) with κ = 1 satisfy

∞ X 1 τ(A(ξ)) = h (k), ξ⊗ki. (6.15) k! κ k=0

Then, for each n ∈ N,

n X n P (n)(ω) = (k) S(n−k)(ω), ω ∈ D0. (6.16) k κ k=0 Proof. By (4.2), (SS3) and (6.15),

∞ ∞ ! ∞ ! X 1 X 1 X 1 hP (n)(ω), ξ⊗ni = h (k), ξ⊗ki hS(n)(ω), ξ⊗ni , n! k! κ n! n=0 k=0 n=0 which implies (6.16).

(n) ∞ Corollary 6.7. Under the conditions of Theorem 6.2, assume that (S )n=0 is a Shef- (n) ∞ (n) ∞ (n) fer sequence. Then, (S )n=0 = (P )n=0 if and only if S (0) = 0 for each n ∈ N.

Proof. By (SS4), for each n ∈ N and ω ∈ D0, we have

n X n S(n)(ω) = P (n)(ω) + S(k)(0) P (n−k)(ω). k k=1 From here the statement follows. Let C(D0) denote the cylinder σ-algebra on D0, i.e., the minimal σ-algebra on D0 with respect to which each monomial h·, ξi (ξ ∈ D) is measurable. A probability measure µ on (D0, C(D0)) is said to have finite moments if, for any n ∈ N and f (n) ∈ D n, Z |hω⊗n, f (n)i| dµ(ω) < ∞. D0 We now propose the following definition. Definition 6.8. Let µ be a probability measure on (D0, C(D0)) that has finite moments. (n) ∞ 0 (n) ∞ Let (S )n=0 be a polynomial sequence on D . The polynomials (S )n=0 are said to be (m) m (n) n orthogonal with respect to µ if for any m, n ∈ N0, m 6= n, f ∈ D , and g ∈ D , Z hS(m)(ω), f (m)ihS(n)(ω), g(n)i dµ(ω) = 0. D0

39 Corollary 6.9. Let µ be a probability measure on (D0, C(D0)) that has ﬁnite moments. (n) ∞ 0 (n) ∞ Let (S )n=0 be a Sheﬀer sequence on D . Assume that (S )n=0 are orthogonal with respect to µ. Then the corresponding operator T has the following representation: Z (TP )(ω) = P (ω + ζ) dµ(ζ),P ∈ P(D0). (6.17) D0

Furthermore, for f (n) ∈ D n, n ∈ N, we have Z hτ (n), f (n)i = hω⊗n, f (n)i dµ(ω). (6.18) D0

P∞ 1 (n) ⊗n Here, τ(ξ) = 1 + n=1 n! hτ , ξ i ∈ S(D, R) is as in (6.3). Remark 6.10. Formula (6.18) states that each τ (n) is the nth moment of the orthogonality measure µ.

Proof of Corollary 6.9. Note that formula (6.17) holds for P = 1. Now, for each n ∈ N and ξ ∈ D, we obtain by (SS4):

n Z X n Z hS(n)(ω + ζ), ξ⊗ni dµ(ζ) = hP (n−k)(ω), ξ⊗(n−k)i hS(k)(ζ), ξ⊗ki dµ(ζ) 0 k 0 D k=0 D = hP (n)(ω), ξ⊗ni = T hS(n), ξ⊗ni(ω).

Formula (6.18) follows immediately from (6.8) and (6.17). Remark 6.11. Note that, in the proof of Corollary 6.9, we only use the fact that R (n) (n) D0 hS (ω), f i dµ(ω) = 0 for all n ∈ N. Corollary 6.12. Assume that the conditions of Corollary 6.9 are satisﬁed. Assume that there exists O ⊂ D, an open neighborhood of zero in D, such that, for all ξ ∈ O, Z exp |hω, ξi| dµ(ω) < ∞. D0 Then, for all ξ ∈ O, Z τ(ξ) = exp hω, ξi dµ(ω), (6.19) D0 i.e., τ(ξ) is the Laplace transform of the measure µ.

Proof. Formula (6.19) follows from (6.18) and the dominated convergence theorem.

Recall that a Sheﬀer sequence on R whose delta operator is the operator of diﬀer- entiation is called an Appell sequence on R.

40 (n) ∞ Deﬁnition 6.13. Let (S )n=0 be a Sheﬀer sequence for the family of delta operators (n) ∞ 0 (Q(ζ))ζ∈D0 = (D(ζ))ζ∈D0 . Then we call (S )n=0 an Appell sequence on D . By (4.2), A(ξ) = ξ in the case of an Appell sequence. Hence, an Appell sequence (n) ∞ (S )n=0 has the generating function ∞ X 1 hS(n)(ω), ξ⊗ni = exp hω, ξi − C(ξ), ω ∈ D0, ξ ∈ D. n! n=0

7 Lifting of Sheffer sequences on R We can extend the procedure described in Section5 to a lifting of Sheffer sequences ∞ on R. Let (sn)n=0 be a Sheffer sequence of monic polynomials on R for the delta operator Q, i.e., Qsn = nsn−1 for each n ∈ N0. Thus, Q has representation (5.1) and ∞ the polynomial sequence (sn)n=0 has the generating function ∞ X un s (t) = exp[ta(u) − c(a(u))], n! n n=0 where a(u) ∈ S(R, R) is given by (5.3) (being the compositional inverse of the q given by (5.2)) and ∞ X k c(u) = cku ∈ S(R, R). k=1

We now consider the family of delta operators, (Q(ζ))ζ∈D0 , given by (5.9). Then A(ξ) is given by (5.10). Furthermore, for k ∈ N, we deﬁne C(k) ∈ D0 k by

(k) (k) (k) (k) k hC , f i := ckhDkf i, f ∈ D . R Here, the operator k is deﬁned by (5.5) and for ξ ∈ D, we denote hξi := d ξ(x) dx. D R Thus, for ξ ∈ D, we deﬁne

∞ ∞ X (k) ⊗k X k C(ξ) := hC , ξ i = ckhξ i =: hc(ξ)i. k=1 k=1

(n) ∞ We now consider the Sheﬀer sequence (S )n=0 for the family of delta operators (Q(ζ))ζ∈D0 that has the generating function

∞ X 1 hS(n)(ω), ξ⊗ni = exp hω, a(ξ)i − hc(a(ξ))i, ω ∈ D0, ξ ∈ D, (7.1) n! n=0

(n) ∞ see (5.4) and Corollary 6.4. Thus, we may think of the Sheﬀer sequence (S )n=0 on 0 ∞ D as the lifting of the Sheﬀer sequence (sn)n=0 on R.

41 (n) ∞ Proposition 7.1. Let (S )n=0 be a Sheﬀer sequence with generating function (7.1). Then, for each n ∈ N, ρ(n) := S(n)(0) ∈ D0 n satisﬁes

(n) ⊗n X Y |B| hρ , ξ i = λ|B|hξ i, ξ ∈ D, (7.2) π∈P(n) B∈π where λk ∈ R are deﬁned by

∞ X λk λ(u) := −c(a(u)) = uk, u ∈ . (7.3) k! R k=1 Furthermore,

n X n hS(n)(ω), ξ⊗ni = hρ(k), ξ⊗kihP (n−k)(ω), ξ⊗(n−k)i, ω ∈ D0, ξ ∈ D, (7.4) k k=0 where hP n(ω), ξ⊗ni is given by formula (5.13).

Proof. By (7.1) with ω = 0 and (7.3), we have

∞ " ∞ # X 1 X λk hρ(n), ξ⊗ni = exp hξki . (7.5) n! k! n=0 k=1 Using Fa`adi Bruno’s formula for the nth derivative of composition of functions, we deduce (7.2) from (7.5). Formula (7.4) immediately follows from (SS4) with ω = 0 (or (SS5)), (5.11), and Proposition 5.1. We can also write down the result of Proposition 7.1 in the following form. By a marked partition of the set {1, 2, . . . , n} we will mean a pair (π, mπ) in which π = {B1,B2,...,Bk} ∈ P(n) and mπ : π → {−, +}. (The value mπ(Bi) ∈ {−, +} may be interpreted as the mark of the element Bi of the partition π). We will denote by MP(n) the collection of all marked partitions of {1, 2, . . . , n}.

(n) ∞ Corollary 7.2. Let (S )n=0 be a Sheﬀer sequence with generating function (7.1). Then, for each n ∈ N, ω, ∈ D0, and ξ ∈ D,     (n) ⊗n X Y |B| Y |B| hS (ω), ξ i =  α|B|hω, ξ i  λ|B|hξ i ,

(π,mπ)∈MP(n) B∈π: mπ(B)=+ B∈π: mπ(B)=−

see formula (5.12) for the deﬁnition of αk. Proof. Immediate from Proposition 7.1.

42 Using Proposition 7.1, we can now immediately extend Corollary 5.2 to the case of a lifted Sheffer sequence. (n) ∞ Corollary 7.3. Let (S )n=0 be a Sheffer sequence with generating function (7.1). d (n) d n Then, for each η ∈ M(R ) and n ∈ N, we have S (η) ∈ Msym((R ) ). Furthermore, d for each Λ ∈ B0(R ), and n ∈ N, we have (n) n S (η))(Λ ) =s ñ(η(Λ)), (7.6)

∞ where (˜sn)n=0 is the Sheffer sequence on R with generating function ∞ X un s˜ (t) = exp ta(u) − vol(Λ)c(a(u)), n! n n=0 R ∞ ∞ where vol(Λ) := Λ dx. In particular, (˜sn)n=0 = (sn)n=0 if vol(Λ) = 1. Proposition 7.4. The statement of Proposition 5.3 remains true for a Sheffer sequence (n) ∞ (S )n=0 with generating function (7.1). Proof. Analogously to the proof of Proposition 5.3, we note that, for any ξ, φ ∈ D satisfying (5.17), exp hω, a(ξ + φ)i − hc(a(ξ + φ))i = exp hω, a(ξ)i − hc(a(ξ))i exp hω, a(φ)i − hc(a(φ))i. The rest of the proof is similar to that of Proposition 5.3. We will now consider examples of lifted Sheffer sequences.

7.1 Hermite polynomials on D0

∞ The sequence of the Hermite polynomials on R,(sn)n=0, is the Appell sequence on R u2 with c(u) = 2 . The Hermite polynomials are orthogonal with respect to the standard ∞ Gaussian (normal) distribution on R. The lifting of (sn)n=0 is the sequence of Hermite 0 (n) ∞ polynomials on D ,(S )n=0, that has the generating function ∞ 2 X 1 (n) ⊗n hξ i 1 2 hS (ω), ξ i = exp hω, ξi − = exp hω, ξi − kξk 2 d . (7.7) n! 2 2 L (R ,dx) n=0

Remark 7.5. For each ξ ∈ D with kξkL2(Rd,dx) = 1, we get from (7.7) that (n) ⊗n hS (ω), ξ i = sn(hω, ξi). (7.8) We see that the Hermite polynomials, S(n), do not actually make use of the spatial structure of the underlying space, D0, but essentially use only the Hilbert space structure of L2(Rd, dx). Formula (7.8) is an exceptional property of the inﬁnite-dimensional Hermite polynomials, compare with the general case discussed in Corollary 7.3.

43 Using either Proposition 7.1 or Corollary 7.2, we easily get an explicit formula

[ n ] X2 n (2k)! hS(n)(ω), ξ⊗ni = (−hξ2i)khω, ξin−2k, 2k k! 2k k=0

n n (2k)! where [ 2 ] denotes the largest integer ≤ 2 . (Note that k! 2k is the number of all partitions π ∈ P(2k) such that each set from the partition π has precisely two elements.) Let µ be the probability measure on D0 that has Fourier transform Z 1 2 exp[ihω, ξi] dµ(ω) = exp − kξk 2 d , ξ ∈ D. L (R ,dx) D0 2 The measure µ is called the Gaussian white noise measure. The Hermite polynomials (n) ∞ (m) (S )n=0 are orthogonal with respect to µ, and furthermore, for any m, n ∈ N, f ∈ D m, and g(n) ∈ D n, Z (m) (m) (n) (n) (m) (n) hS (ω), f ihS (ω), g i dµ(ω) = δm,nn!(f , g )L2(Rd,dx) n , (7.9) D0 see e.g. [6,15]. As pointed out in the Introduction, the inﬁnite dimensional Hermite polynomials are well-known and play a fundamental role in Gaussian white noise analysis, see e.g. [6, 15,16,31] and the references therein. In white noise analysis, one usually writes : ω⊗n : for S(n)(ω) and call it the nth Wick power of ω. In that context, the transformation T given by formula (6.17) is known as the C-transform.

7.2 Charlier polynomials on D0

∞ The sequence of the Charlier polynomials on R,(sn)n=0, is the Sheﬀer sequence with a(u) = log(1 + u) and c(u) = eu − 1, so that λ(u) = −u. The Charlier polynomials are orthogonal with respect to the Poisson distribution corresponding to the intensity ∞ 0 parameter 1. The lifting of (sn)n=0 is the sequence of the Charlier polynomials on D , (n) ∞ (S )n=0, that has the generating function

∞ X 1 hS(n)(ω), ξ⊗ni = exp [hω, log(1 + ξ)i − hξi] . n! n=0

∞ 0 Note that the corresponding binomial sequence is ((ω)n)n=0, the falling factorials on D . By Proposition 7.1,

n X n hS(n)(ω), ξ⊗ni = h−ξikh(ω) , ξ⊗(n−k)i. (7.10) k n−k k=0

44 Furthermore, by Corollary 6.6, we obtain

n X n h(ω) , ξ⊗ni = hξikhS(n−k)(ω), ξ⊗(n−k)i. (7.11) n k k=0 Compare formulas (7.10) and (7.11) with Corollaries 2.9 and 2.10 in [20], respectively. Note that the latter results were obtained only for ω from the conﬁguration space Γ. Let µ be the probability measure on D0 that has Fourier transform Z Z exp[ihω, ξi] dµ(ω) = exp (eiξ(x) − 1) dx , ξ ∈ D. D0 Rd The measure µ is concentrated on the conﬁguration space Γ and is called the Poisson (n) ∞ point process, or the Poisson white noise measure. The Charlier polynomials (S )n=0 are orthogonal with respect to the Poisson point process µ and formula (7.9) holds true in this case. (n) ∞ The Charlier polynomials (S )n=0 play a fundamental role in Poisson analysis, see e.g. [17, 19, 22]. In this analysis, the transformation T given by formula (6.17) is also known as the C-transform.

7.3 Orthogonal Laguerre polynomials on D0 It follows from (5.29) that, for each parameter k > −1, the sequence of the Laguerre [k] ∞ polynomials (pn )n=0 on R corresponding to the parameter k is a Sheﬀer sequence, ∞ [−1] ∞ whose corresponding binomial sequence is (pn)n=0 = (pn )n=0, see (5.30). For each [k] ∞ k > −1, the Laguerre polynomials (pn )n=0 are orthogonal with respect to the gamma distribution 1 χ (t)tke−t dt. Γ(k + 1) (0,∞)

∞ [0] ∞ In particular, for k = 0, the Laguerre polynomials (sn)n=0 := (pn )n=0 are orthogonal −t ∞ with respect to the exponential distribution χ(0,∞)(t)e dt on R. By (5.29), (sn)n=0 u is the Sheﬀer sequence with a(u) = 1+u and c(u) = − log(1 − u), so that λ(u) = − log(1 + u). ∞ (n) ∞ 0 The lifting of (sn)n=0 is the sequence (S )n=0 of the Laguerre polynomials on D that has the generating function

∞ X 1 ξ hS(n)(ω), ξ⊗ni = exp ω, − hlog(1 + ξ)i . (7.12) n! 1 + ξ n=0 Note that the corresponding polynomial sequence of binomial type is the Laguerre (n) ∞ sequence (P )n=0 with generating function (5.31), see Subsection 5.4. Analogously to formula (5.32), we will now present a combinatorial formula for hS(n)(ω), ξ⊗ni.

45 We can identify each permutation π ∈ S(n) with c(π) := {ν1, . . . , νk}, the set of the cycles in π. For each cycle νi ∈ c(π), we denote by |νi| the length of the cycle νi. We deﬁne MS(n) := (π, mπ) | π ∈ S(n), mπ : c(π) → {+, −} , compare with the deﬁnition of MP(n) above. Note that, for a given subset of N that has m elements, there are (m − 1)! cycles of length m that contain the points from this set. Hence, by Corollary 7.2 and (7.12), we get:     (n) ⊗n X Y |ν| Y |ν| hS (ω), ξ i =  |ν| − ω, (−ξ)   (−ξ)  .

(π,mπ)∈MS(n) ν∈c(π): mπ(ν)=+ ν∈c(π): mπ(ν)=−

∞ [k] ∞ By Corollary 7.3, formula (7.6) holds with (˜sn)n=0 = (pn )n=0, the Laguerre polynomials on R corresponding to the parameter k = vol(Λ) − 1 > −1. Let µ be the probability measure on D0 that has the Laplace transform Z Z e−hω,ξi dµ(ω) = exp − log(1 + ξ(x)) dx , ξ ∈ D, ξ > −1. D0 Rd The µ is called the gamma measure, or the gamma completely random measure. It is P d concentrated on the set of all (positive) discrete Radon measures η = i siδxi ∈ M(R ) with si > 0 for all i. Note that, with µ-probability one, the set of atoms of η, {xi}, is dense in Rd. (n) ∞ As follows from [21, 22], the Laguerre polynomials (S )n=0 are orthogonal with respect to the gamma measure µ, and furthermore, for any ξ, ψ ∈ D and m, n ∈ N, Z (m) ⊗m (n) ⊗n X Y |ν| hS (ω), ξ ihS (ω), ψ i dµ(ω) = δm,nn! (ξψ) . 0 D π∈S(n) ν∈c(π)

(n) ∞ The Laguerre polynomials (S )n=0 play a fundamental role in gamma analysis, see e.g. [21,22,25,26].

Acknowledgments The authors acknowledge the financial support of the SFB 701 “Spectral structures and topological methods in mathematics”, Bielefeld University. MJO was supported by the Portuguese national funds through FCT—Funda¸cãopara a Ciênciae a Tecnolo- gia, within the project UID/MAT/04561/2013. YK and DF were supported by the European Commission under the project STREVCOMS PIRSES-2013-612669.

46 Appendix: Formal tensor power series

We ﬁx a general Gel’fand triple (2.1). The following proposition is a direct consequence of formula (2.9).

P∞ (n) ⊗n (0) Proposition A.1. Let F (ξ) = n=0hF , ξ i ∈ S(Φ, R) be such that F (0) = F 6= P∞ (n) ⊗n 0. Then there exists a unique n=0hG , ξ i ∈ S(Φ, R) such that

∞ ! ∞ ! X X hF (n), ξ⊗ni hG(n), ξ⊗ni = 1. n=0 n=0

Explicitly, G(0) = 1/F (0) and for n ≥ 1, G(n) is recursively given by

n−1 1 X G(n) = − F (n−i) G(i). F (0) i=0 We will denote ∞ !−1 ∞ X X hF (n), ξ⊗ni := hG(n), ξ⊗ni. n=0 n=0

Remark A.2. It follows from Proposition A.1 that, for any F (ξ),G(ξ) ∈ S(Φ, R) with F (0) 6= 0, we obtain G(ξ) ∈ S(Φ, ). F (ξ) R P∞ n P∞ (n) ⊗n Deﬁnition A.3. Let R(t) = n=0 rnt ∈ S(R, R). For each F (ξ) = n=1hF , ξ i ∈ S(Φ, R) with F (0) = 0, we deﬁne a composition of R and F , denoted by R ◦ F (ξ) or

∞ ∞ !n X X (k) ⊗k R(F (ξ)) = rn hF , ξ i , n=0 k=1

P∞ (n) ⊗n (0) as the formal series G(ξ) = n=0hG , ξ i ∈ S(Φ, R) with G := r0 and

n (n) X X (k1) (k2) (km) G := rm F F · · · F , n ∈ N. m m=1 (k1,...,km)∈N k1+···+km=n

P∞ ⊗n P∞ ⊗n Deﬁnition A.4. Let A(ξ) = n=1 Anξ ,B(ξ) = n=1 Bnξ ∈ S(Φ, Φ). We deﬁne a composition of A and B, denoted by A ◦ B(ξ) or

∞ ∞ !⊗n X X ⊗k A(B(ξ)) = An Bkξ , n=1 k=1

47 P∞ ⊗n as the formal series C(ξ) = n=1 Cnξ ∈ S(Φ, Φ) with

n X X Cn := Am Bk1 Bk2 · · · Bkm , n ∈ N. (A.1) m m=1 (k1,...,km)∈N k1+···+km=n

m Here, for (k1, . . . , km) ∈ N with k1 + ··· + km = n, we denote

Bk1 Bk2 · · · Bkm := Symn(Bk1 ⊗ Bk2 ⊗ · · · ⊗ Bkm ),

⊗n n where Symn ∈ L(Φ , Φ ) is the operator of symmetrization, see (2.2). Similarly to Definitions A.3 and A.4, we give the following P∞ (n) ⊗n P∞ ⊗n Definition A.5. Let F (ξ) = n=0hF , ξ i ∈ S(Φ, R) and A(ξ) = n=1 Anξ ∈ S(Φ, Φ). We define a composition of F and A, denoted by F ◦ A(ξ) or

∞ * ∞ !⊗n+ X (n) X ⊗k F (A(ξ)) = F , Akξ , n=0 k=1

P∞ (n) ⊗n (0) (0) as the formal series G(ξ) = n=0hG , ξ i ∈ S(Φ, R) with G := F and

n (n) X X ∗ ∗ ∗ (m) G := (Ak1 Ak2 · · · Akm )F , n ∈ N. m m=1 (k1,...,km)∈N k1+···+km=n

∗ 0 0 k Here Ak ∈ L(Φ , Φ ) is the adjoint of Ak. Remark A.6. It follows from Deﬁnition A.5 that

F (A(ξ)) = F (0)   ∞ n X X X (m) ⊗k1 ⊗k2 ⊗km  +  F , (Ak1 ξ ) (Ak2 ξ ) · · · (Akm ξ )  .  m  n=1 m=1 (k1,...,km)∈N k1+···+km=n

Proposition A.7. Let F (ξ) ∈ S(Φ, R) and let A(ξ),B(ξ) ∈ S(Φ, Φ). Then (F ◦ A) ◦ B(ξ) = F ◦ (A ◦ B)(ξ), the equality in S(Φ, R). Proof. The proposition follows from Deﬁnitions A.4 and A.5, see also Remark A.6. We leave the details to the interested reader.

48 Remark A.8. In view of Proposition A.7, we may just write F ◦ A ◦ B(ξ). As easily seen, a similar statement also holds for the composition S ◦ R ◦ F (ξ) ∈ S(Φ, R), where S,R ∈ S(R, R) and F ∈ S(Φ, R), and for the composition A ◦ B ◦ C(ξ), where A, B, C ∈ S(Φ, Φ). Deﬁnition A.9. Let A(ξ) ∈ S(Φ, Φ). Then B(ξ) ∈ S(Φ, Φ) is called the compositional inverse of A(ξ) if A ◦ B(ξ) = B ◦ A(ξ) = ξ. Remark A.10. Note that, if B(ξ) ∈ S(Φ, Φ) is the compositional inverse of A(ξ) ∈ S(Φ, Φ), then A(ξ) is the compositional inverse of B(ξ).

P∞ ⊗n Proposition A.11. Let A(ξ) = n=1 Anξ ∈ S(Φ, Φ) with A1 ∈ L(Φ) being a homeomorphism. Then there exists a unique compositional inverse B(ξ) of A(ξ).

P∞ ⊗n Proof. Let us ﬁrst prove that there exists a unique B(ξ) := n=1 Bnξ ∈ S(Φ, Φ) such that A ◦ B(ξ) = ξ. It follows from formula (A.1) that C1 = A1B1. Hence, for −1 C1 = 1, we must have B1 = A1 . Now, by (A.1) for n ≥ 2, we get

n X X Cn = A1Bn + Am Bk1 Bk2 · · · Bkm . m m=2 (k1,...,km)∈N k1+···+km=n

Hence, we get Cn=0 for n ≥ 2 if and only if

n −1 X X Bn = −A1 Am Bk1 Bk2 · · · Bkm . m m=2 (k1,...,km)∈N k1+···+km=n

˜ P∞ ˜ ⊗n Similarly, we prove that there exists a unique B(ξ) := n=1 Bnξ ∈ S(Φ, Φ) such ˜ ˜ −1 that B ◦ A(ξ) = ξ. Here B1 = A1 and for n ≥ 2,

n−1 ˜ X X ˜ −1 ⊗n Bn = − Bm Ak1 Ak2 · · · Akm (A1 ) . m m=1 (k1,...,km)∈N k1+···+km=n

Finally, we prove that B(ξ) = B˜(ξ). Indeed, we get, using Remark A.8,

B˜(ξ) = B˜ ◦ (A ◦ B)(ξ) = (B˜ ◦ A) ◦ B(ξ) = B(ξ).

Hence, the proposition is proven.

P∞ ⊗n Remark A.12. It follows from Proposition A.11 and its proof that, if A(ξ) = n=1 Anξ ∈ P∞ ⊗n S(Φ, Φ) with A1 = 1, then its compositional inverse B(ξ) := n=1 Bnξ exists and B1 = 1.

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